I don't see how you could pay hers and not pay it for all of your other kids. You can't treat one person different than the others. How many kids do you keep? That could get quite expensive. If the contract doesn't call for you to pay, you really can't do it and stay afloat. I would just try to explain that to her and if she won't listen, she will just have to do whatever she feels necessary.
The only problem I see is if you are obligated because you are receiving payment from the state but even if that is so, you would certainly only be obligated to refund whatever the state paid you for her child, not the amount that she paid to the other daycare.
Good luck to you.
