Can a babysitter that I terminated for not properly caring for my son take me to court to pay her for the days she did not work and the days in our contract?

There are two issues

First, did the contract you mention guaranty or protect her employment, state that she could only be fired "for cause," or otherwise state that her employment was *not* employment at will? If it did not, then you could fire her at any time, for any reason, and she would have no claim against you--employment is normally employment at will unless a contract makes it otherwise, and when employment is employment at will, it may be terminated without warning or notice, for any reason whatsoever.

If the contract did state or imply that her employment could only be terminated "for cause," then the second question comes up: did you have good cause? You say she was not properly taking care of your son--what does that mean? If she was not keeping him safe, or was neglecting him him, or not following your specific instructions, then that should constitute valid cause, and if fired for cause in this case, there'd be no liability. However, if you simply did not like certain things she did without giving her specific instructions to the contrary (since violating an employer's instructions is cause for termation), then it might be that if she could only be fired for cause that you breached the contract and could be liable for what she should have been paid under it. For example, say she was letting your son watch TV and you discovered that and fired her for it--unless you had told her "no TV," that would most likely not be sufficient cause to fire her, if under her contract, she could only be fired "for cause."