Here's the thing. She may turn this over to a collection agency but chances are she'll never take it to court. She terminated you, she broke the contract therefore you do not owe her anything beyond the days she watched your child.
I would ignore it or simply tell the collection agency when they call that she broke the contract and you have the letter to prove it. Would they like to see it? Then tell them to not contact you anymore the debt is a false claim. If she does spend the money to take you to court you will just need to present the letter and explain that services were terminated by the facility therefore she broke the contract and you do not owe any money.
If she did carry this far enough to put this debt on your credit report, which can have an explanation tagged on it to dismiss it totally, I would simply let it go still. People have stuff on their credit report all the time that is wrong. A simple note attached to the claim is all that is needed for anyone looking to dismiss it.
I can tell you from 13 years experience in child care that she would be stupid to take you to court. Most "contracts" in child care are not worth the paper they are written on. They are not legally binding contracts. The provider will tell you they are but my attorney advised me that unless they are notarized and filed properly they are not legal documents. They are merely an agreement between two people that lists out what the other one wants the other one to do. That's it.
I stopped even writing one up. I did have my parents sign a paper that said they had read the parent handbook and agreed to the terms of it. That they understood about the late fees and that they must be paid before their child can come back. That payments were due on Monday mornings and that if they did not pay it their child could not stay until they did. I did make exceptions of course for parents who were good parents that usually paid on time and didn't make unreasonable demands.
If you think talking to this person is something that can be worked out then by all means go ahead and try but do NOT get sucked in and think she has any real intentions of taking you to court. I would not even pay this any attention at all. In fact I would file it away and ignore it.
If I did get a call from a debt collector I would explain it was a false claim and they needed to take it up with her. Then I would contact my attorney and ask them to write her a letter stating that if she did not cease from contacting his client and passing false information to debt collectors, and basically committing slander against you that he would file XXX papers against her immediately.
Follow through with the call to licensing. Have you looked at their online inspection reports? Do they have many they didn't pass 100%? I always had something, the door was not locked the way they wanted, the toilet seat was not down on the toilet in the school kids bathroom, the band aids in the first aid kit were getting low. stuff that is important but not life threatening.
I would tell them everything that I noticed. I am adding some links that will have some licensing information so that perhaps you can brush up on the things you have observed she is not doing or does not enforce.
Please note that she may have sent the bill before terminating your services or by a billing person. This is unlikely but still. She broke the contract by terminating your services without the 2 weeks notice she is asking you to pay for.
A variety of different links within the CA child care licensing area. I would click on it then decide which ones to research.