While regularly used in conversation, I'd never use it in any written paper.
Ask her if that's how she would write it should she be writing a formal document.
Spoken language is generally more flexible, and I'm wondering if she doesn't mean "I've done my homework" and due to accent it sounds like "I'm", because "I've done my homework" would be proper English.
Edit to add more information
"I did my homework" is correct using simple past form of "do".
"I have done my homework" is correct using present perfect tense form of "do".
"I am done with my home work" (or "I'm", being a contraction of I am) is using "done" as an adjective, the verb is am (first-person singular present indicative of be).
"I'm done my homework" is just wrong. Done is being used as an adjective (I am done), and "my homework" is an incomplete clause.
Either you must remove the verb am and replace "done" with "did", or add the word "with" to link "my homework" to the statement.
The only way this could possibly be correct, is if people are pronouncing the "ve" of "I've done my homework" as an "m" for some odd reason. "I've done my homework" is proper (I have done my homework).
On an unrelated note, I agree Jay's answer philosophically.
A.1.c In modern use (with following
dative) often = ‘such as’, introducing
a particular example of a class
respecting which something is
1886 Stevenson Lett. (1899) II. 41
A critic like you is one who fights
the good fight, contending with
stupidity. 1887 Colvin Keats i. 1 A
birth like that of Keats presents to
the ordinary mind a striking instance
of nature's inscrutability.
So there is essentially no difference between "like" and "such as" when used as you described. The two have been interchangeable since the late 1800s.