It depends on what's in your contract, I think. Do you have a provision that they have to pay for those last two weeks of care after you give them a two weeks notice, whether the child attends or not? If so, you can wave the contract at her and tell her she is legally obligated to stick to the terms.
The thing is, this lady sounds like she could get quite nasty if she wants to, and I have a feeling that getting money from her is going to be a battle all the way. I think you are better off without her, I just hope she will give you what you are due.

ETA I would definitely give it my all. The friendship is a bust anyway, she sounds like she'd gladly walk all over you to get what she wants, so I would definitely pursue that money.