The Snow Queen is just lonely and wants friends.
The Snow Queen was trying to help Kai the whole time
That's why she agreed to let him go if he solved the puzzle. Solving it required he remember what beauty was again. it also explains why there's no indication of her being upset about him leaving.- Besides her attitude could explain that she is just a serious doctor who is healing Kai's devil mirror pieces and quarantines him so he cannot hurt anyone? Solving the puzzle maybe is the way for making sure he gets his humanity again.
