"It seems I broke your player."
Translation:Кажется, я сломала твой плеер.
There is a slight difference between "кажется" и "мне кажется". As I see, both variants "Мне кажется, я сломал..." and "Мне кажется, ЧТО я сломал..." are natural, but it seems "Кажется, ЧТО я сломал..." is quite unnatural though "Кажется, я сломал..." is still a good variant. That's a pretty hard thing to describe, but from these four variants three are really good and one is a bit strange and no one will say it that way. Unfortunately, that variant is your variant.
Разбил is kind of more specific -- smashed, crashed, shattered. You would probably say that if you dropped the thing or hit it so it is now physically damaged and has cracks, or parts of it broke off. When we say сломал it can mean anything that resulted in it not working properly or not working at all, and it can be with or without any visible physical damage.
Okay. I did use the word broke, but I never said impossible. Of course it is possible, I am just saying that there is some difference between those two things. What I mean is that if you don't have any context and don't know for sure what exactly happened and in what way the thing is damaged, you should better say "сломал" rather then "разбил", because it's more abstract.