First, if you're actually teaching English to non-native speakers, you must learn and use at least those IPA symbols that represent English phonemes. Get yourself a copy of Kenyon and Knott and use it; or borrow one of your students' bilingual dictionaries.
If you help them, your students can understand the pronunciations as they appear in their bilingual dictionaries, which always use IPA symbols; but they won't understand what you mean, otherwise. And they won't improve.
Second, the correct vowel symbol is /ɪ/, which appears in hill, shit, think, Mrs, and rim. We're talking about a high front lax vowel here, not a "short vowel", and certainly not about the letter I; calling something "short I" is 18th-Century nonsense. It's not short in Modern English, and it's not pronounced /ay/. How words are spelled in English has very little to do with how they're pronounced. English spelling is a good system for Middle English, but it's a bad system for Modern English.
Third, you're correct that a vowel before a nasal consonant /m, n, ŋ/ in English will sound different, because all vowels (including /ɪ/) that appear before nasals are nasalized. That is to say, they are pronounced with the nasal passage open, instead of closed.
This is because the nasal passage has to open in order to pronounce the nasal consonant, and the vowel anticipates this opening. This is a standard, predictable, utterly automatic, allophonic process in English phonology, and it does make the vowel sound different. But it doesn't make it a different vowel, because nasalized vowels occur in English only before nasal consonants, so they never contrast with non-nasalized vowels.
Finally, it's phonetics, not phonics. Phonics is a system of lying to English-speaking children in an effort to teach them how to spell their own language, and is useless for adults who don't already know English. (It's pretty useless for children, too, as we can see anywhere.)