Oh, she could keep it secret. ... Whether or not she'd use it as leverage, especially if he's actively trying to keep it hidden, that's another story. Probably. And if we have mysterious incidents of microbots in the nighttime the chances of China letting Skulduggery know are pretty high. But apart from that, yes. :P

And she has actually taught before, in-canon, so it's not completely out of the question! I think she'd be a bit picky regarding her students, and would be more suited as a tutor than in offering actual open classes, but that'd be interesting to play with.