Melyiket/melyeket refers to one or more definite objects, so it requires using the definite conjugation, which in this case is marked by an -i suffix of the verb.
I have never heard anyone use the word kinds in this way. Maybe what kinds of.... , or- what kind? What kind of apples...? What kinds of things?
Yeah, it doesn't make a lot of sense without context. Say, you know Zsuzsa is lesbian and you're telling your friend about it: "Zsuzsa likes girls." - "Oh, what kinds (of girls) does she like?" - "Well, she has a faible for blondes. And shortstacks."
You would probably express it a little differently in English, but Hungarians do it just like that.
Since 'milyeneket' is plural, wouldn't the proper translation be "What kinds (pl.) does Zsuzsa like?
Yes, that should be the case. It used to be "kinds".
EDIT: Wait, not necessarily. Milyenek is talking about multiple objects, but they don't have to be of different kinds. So both answers are alright.