This answer is based on a question migrated from here to Japanese Language and Usage a while back.
Below I've pasted the old answer in its entirety, but I want to address some things in the newly-worded question. Your concern with Japanese students' spelling lies primarily in the fact that they are writing in romaji, and not in English. You have to drill into their heads this idea that romaji does not mean English. No English speaker who doesn't understand Japanese sytems of romanization would look at "geisya" and know to pronounce "sha," and that should be your pedagogical basis. As someone who teaches Japanese students English I cannot emphasize enough that romaji does not mean English. That said, I think you can teach them the modified Hepburn style, though you don't need to call it Hepburn. Just say "し" sounds like "shi" in English, so that's how we write it. "しゃ sounds like 'sha.'" Make sure they understand that "sya" as romanization (that is, kunrei-shiki) is formed by し (s) + ゃ (ya), and that because this romanization is based on kana orthography it will not be understood in English.
In short, to summarize those rules, you'll be using Hepburn without macrons or elongated vowel sounds. Some exceptions apply (as detailed below), but if they're at the level where they're just learning how to spell I think we can just tell them to learn the exceptions later or as they come across them in regular use. I think one of the bigger issues you'll find Japanese students have is that they can't even spell non-Japanese English words correctly and will try to write based on romaji anyway. That's a whole other can of worms, though.
Here is my answer from JLU in full:
Generally speaking in English words will be spelled in a way that resembles Hepburn romanization but without macrons or any special characters, instead just writing it 'plain.' There are not that many words in regular use in English that are of Japanese origin, but those that are do follow patterns (with a few exceptions).
For example, we say Tokyo, Kyoto, and Osaka, with no macrons and no doubling of vowels (No Toukyou, Kyouto, or Oosaka). The other features of Hepburn are present, though. So for example しゅ will be "shu" (not syu), as in Kyushu. じゃ will be ja, as in ninja (cf. ninjya). しゃ is "sha" as in shamisen, し will be shi, as in sushi. つ will be tsu, as in tsunami, and づ "dzu" as in kudzu. ふ will be fu, as in futon. There are many other words that follow the patterns as expected and that don't approach any touchy areas of romanization.
There are some exceptions, though. For example 能 (theater) is spelled as "Noh," which breaks from the patterns and uses the h to add emphasis to the longness of the o sound. Shiitake maintains the double i. Sometimes ん is written using an "m" instead of "n" as in tempura. You also have 浮世絵 which gets hyphenated, as in ukiyo-e.
There are some other words that are from Japanese but underwent complete spelling changes rather than pure romanization, like 大君 (tycoon) and 力車 (rickshaw).
So yeah, the correct English spellings usually follow Hepburn pretty closely, but there are exceptions. As forms of romanization, Kyusyu and the like aren't wrong, but if we're writing in English then I would say those should be corrected to writings that are understandable to English speakers, so definitely Kyushu if it's English.
If you want to peruse a list, you can take a look at the Wikipedia article List of English words of Japanese origin.
Also, just to quickly address a comment that snailplane made about place names, the rule applies the same ways but with occasional exceptions. For example we write "Niigata" instead of "Nigata," and in fact I see even the smallest, most obscurely-named villages get macron-free Hepburn romanization on road signs and stuff.
Another very small thing to note is that we don't usually use an apostrophe to denote the division between "んい", as in Kenichi/Ken'ichi (けんいち, with the former being the more common).
The short answer is that each word has its own history (also called its etymology) that traces where the word came from. The same is true of letters.
The word quick, for example, has in fact been spelled with a c and k in its distant past. From etymonline.com:
Old English cwic "living, alive, animate," and figuratively, of mental qualities, "rapid, ready," from Proto-Germanic *kwikwaz (cf. Old Saxon and Old Frisian quik, Old Norse kvikr "living, alive," Dutch kwik "lively, bright, sprightly," Old High German quec "lively," German keck "bold"), from PIE root *gweie- "to live" (see bio-).
How we settled on quick instead of kwick is anybody's guess, but it's worth pointing out that very few modern English words begin with "kw-". Somewhere along the line, "qu-" became the dominant form. But your same curious question could be asked about words that begin with the Z sound, yet start with an X, like xenon and xylophone.
You can also read more about the letter Q from the same website. I suggest you and your daughter have a look.
Best Answer
The best way to explain why the same letter patterns are pronounced differently in different words is just to explain the actual historical reasons for it. You'll have to choose the timing and the amount of detail to suit your students, but here are the main facts:
The Latin alphabet had about 23 letters but Anglo-Saxon and Middle English had about 45 phonemes. This meant that the same letters had to stand for multiple sounds. This especially causes ambiguity with vowels, which is why 'ou' and 'au' are particularly troublesome. Latin had no 'aw', short 'o͝o', or short 'ă' sound, and no letters for them. The two-letter combinations 'sh', 'ch', 'th', and 'dg' represent consonants not found in Latin. Anglo-Saxon writers represented hard and soft 'th' by two runic letters, 'þ' and 'ð', but not very consistently, and they fell out of use shortly after printing began. There's lots more, but you get the idea: too many phonemes and not enough letters due to imposing the Latin alphabet onto English.
Most silent letters stand for sounds that have entirely dropped out of modern English. For example, knight was pronounced /kniçt/. Modern English no longer has /ç/ (like German Ich), and in Modern English, /kn/ can't start a syllable. (A few dialects still retain these sounds, though.) This is why 'gh' is particularly ambiguous: sometimes it was dropped, and sometimes it shifted to /f/.
The use of 'o' to represent short 'ŭ' before 'v', 'n', and 'm', as in oven, ton, and come, developed when 'u' and 'v' were both the same letter. 'vv' and 'vn' were written as four short, vertical strokes—just like 'nn' or 'nu'. 'vm' was five vertical strokes, just like 'mv'. Writing the 'ŭ' as 'o' in these circumstances clarified the writing at the expense of complicating the spelling rules.
Anglo-Saxon and Middle English pronunciation varied a great deal from region to region, so there was no way to make a single spelling for all the regional pronunciations. Indeed there was no standardization of spelling in Middle English. Standard spellings didn't take root until dictionaries started coming out in the 1600s, and not really until Samuel Johnson's dictionary of 1755. Standardization tended to favor the London pronunciation, but occasionally froze one region's pronunciation with another region's spelling; bury is the most famous example.
Silent 'e' was pronounced in Middle English. It's silent in Modern English but still retains two roles in spelling: softening preceding 'c' and 'g', and making the previous vowel long if both stressed and separated from the 'e' by only one consonant. The latter is due to a rule of Anglo-Saxon pronunciation that a stressed vowel is long before one consonant and another vowel, and short before two consonants and another vowel. This is why we have a lot of doubled letters in words of Anglo-Saxon origin: to indicate that the preceding vowel should not go long; for example, ban becomes banned.
The business with soft and hard 'c' and 'g' comes from changes in the pronunciation of Latin that occurred in Europe somewhere around 200–600 A.D. This convention came to English with the Norman invasion of 1066. The Norman invasion also brought English the great majority of its modern vocabulary, and its basically French spellings for all the new words. French had nothing like the aforementioned rule of doubled consonants and short vowels. This is why we write liberty and lateral (French-style spelling) rather than libberty and latteral (Anglo-Saxon-style spelling).
Johnson's dictionary cemented the convention of spelling words to reflect their etymology. Hence the "three spelling systems" in my first attempt at an answer. Johnson goofed on a lot of his etymologies and introduced a number of inadvertent inconsistencies, some of which have stuck and some of which are forgotten. There hasn't been much standardization for spelling words that come from languages other than Anglo-Saxon, Latin, and Greek. Competing standards for Hebrew, for example, result in Chanukah as well as Hanukkah; the situation with Arabic is a catastrophe. But overall, Johnson's approach is the modern approach: choosing spellings to distinguish roots with different meanings while trying to respect old precedents, making reasonable compromises between many competing concerns.
Noah Webster's dictionaries of 1806–41 got rid of a number of irregularities and simplified a number of French-style spellings, resulting in the divide between American and British spelling. Webster adhered to the principle of etymological spelling, following it more consistently than before. For example, he changed the the British/French -ise suffix to the more etymologically faithful -ize.
To summarize: English spelling arose first from a Procrustean fit to an alphabet without enough letters, and accumulated lots of tweaks and compromises to adapt to problems like invaders with a new language and changes in pronunciation. Eventually, a mostly etymology-based spelling system emerged which turned the "more phonemes than letters" problem into a virtue: using the different ways of spelling the same sound to represent morphemes rather than phonemes consistently (except for all those aforementioned irregularities).
Hopefully, as each quirk of spelling comes up, you can provide just enough history so that students can at least understand the reason for it. They might find some reasons disappointing, as in the case of slaughter, and they might find others enlightening, like why improvise doesn't have a 'z'. Either way, understanding the reasons for English spelling makes it a whole lot easier to learn and remember. It should at least clear the feeling of hopelessness that descends when it starts looking like English spelling is nothing but exceptions.