"I am already seventeen years old."
Translation:Мені вже сімнадцять років.
Because "я" is not the subject of this sentence.
The subject is "сімнадцять років". The underlying structure is "Мені вже є сімнадцять років" (skipping "є" sounds more natural in the standard Ukrainian language). Literally it's "To me already [it] is seventeen years" or "To me already [there] are seventeen years", it's hard to represent in English :)
So, the subject is "seventeen years" and the verb is "to be", while "мені" is in Dativ. So "я" has no place there.
I hope it's clear, if not, please ask more :)