Welcome to the forums. The correct spelling is "mühendislik fakültesi" In this case, and usually the -i is used to refer a noun clause. For instance "aynanın camı" which means "the glass of mirror." You can think this -ı or -i or -u or -ü as "of" in English. (-ı, -i, -u, -ü can be all used depending on the noun, it is just an accord of suffixes to nouns) It can also refer to an ownership. For example, "Sevgi'nin kalemi", "the pencil of Sevgi". ("Sevgi" is a Turkish name.)
In your example, the -i is "of", it means "faculty of ..."
However, the -lik or -lık or -luk (depending on the noun" are the suffixes which make nouns to another nouns. For example, "kalem" means "pencil" and it is a noun. If we add "-lik" it becomes another noun which means "pencil case". In your question, "mühendislik" means "being an engineer", and "mühendislik fakültesi" is "Faculty of Engineering".
These suffixes are so difficult but I hope I can explain something.
I understand that -nın or -nun (or its other variants, in accordance with vowel harmony) are suffixed to a noun ending with a vowel. But what about those not ending with a vowel? Do we add anything or not? Can anybody give some examples?
But I am still confused about -lik (or its variants). To give an example, someone asked me whether the room I rented was dönemlik (i.e. whether I paid for a semester only). What did he really mean?