Learn English – /i/ sound before “ng” and “nk”

I'm a substitute teacher and recently was teaching a kindergarten class about long i sound. They were crossing out words without long i, circling words with long i. One of the words was ink. I told them no, listen , we don't say i nk (say it with a long i to see what I mean) and they crossed it out. Later, looking at the teacher's edition, it had ink circled. I thought it was just a mistake then I saw that an ink bottle was actually used as an example in the book for the long i sound. I thought something was terribly wrong so I looked it up in the dictionary—it shows i in ink as a long i sound (I looked at many dictionaries, all were the same). This can't be right, but I'm wondering if it's one if those things that's just been accepted and not questioned. Or if it's a category that hasn't been explored yet as needing a separate sound to clarify, like words with r-controlled vowels. Any comments on this would be extremely helpful.

Edit:

I realized I made a mistake with my original post and used long I all the way through. I meant to say that in ink, think, pink, thing, ring, king, etc., in other words, words ending in "nk" and "ng", the "i" is usually pronounced more like a long e sound, like e in meet. At least, that is how I have always pronounced it and heard it pronounced. I'm from California so this could be regional, but I've never heard it pronounced with a short i like in it. Fumblefingers listed words in which a short i occurs, including ink, pink, bit, fit. I definitely hear short i in bit and fit, sounds the same. In ink and pink, i does not sound the same to me, nor have I heard people say it with the same i sound as in bit. Unless the i is getting so quickly blended into the "ng" that it is almost ignored, in which case it should have a special sound category like we teach r-controlled vowels. The pronunciation rules could be very different between the US and the UK .