ETA
In child care it's pretty common for payment to be weekly even if your child misses days due to sickness or vacation or anything. Most centers will give you 2 weeks each year for full time kids where the parents don't bring the kids and they don't have to pay any funds to keep their child's spot.
So trying to make you pay her for 2 weeks when you are on vacation or not there isn't right, it is unethical completely.
Normally a person would pay for child care 50 weeks of the year. When a parent leaves they don't owe for something that no longer applies to them. You get to not pay her those two weeks of vacation. That's zero income for her if she closes especially because YOU have to pay another person to watch your kids.
YOU get to not pay her for a couple of weeks to go on vacation. You don't pay her for her vacation. Period.
Turn her in to the BBB and write a review on FB if your town has a business review page.
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I have owned my own center. Going in we know that our contracts are pretty useless when it comes right down to it. Sure, it's a signed agreement but if you walk out she is not going to take you to court. She might file something against you but you just have to mention, in your credit report that this is a messed up thing and you don't owe the money.
She's known for months that you're leaving and she's probably got kids lined up to come.
Don't worry about it and let her know that you'll fight it if she tries to take you to court. Tell her "Okay, that's fine with me. I gave you 2 months notice and then you decided I owe for time my children won't even be enrolled for. So sorry. Take me to court, I'd like to have my time in front of the judge."
I think it's more hassle for a director to take time out to go to court and it's stupid for her to ask you to pay money you do not owe.