You cannot use "kogo" here, however it is probably a common grammatical mistake and everyone would understand you. If you want to use "kogo", say: "Do kogo należy to lustro?" - "To whom does this mirror belong".
Czyj, czyja etc. is the question word about posession (eng. whose).
Kogo and czego are used to make questions about object in genitive, but not about posession (kogo słuchasz? czego nie widziałeś?).
But, on the other hand, many Poles do not differentiate this and use kogo in both contexts. Yet, it is considered a mistake by normativists and sounds like poor Polish to many people.