Legal Question in Business Law in California

What does the contract say? Seems a little strange the no longer need you but need you fir two more weeks. The contract would guide in a lawsuit.the worst you would pay are some damages in the costs it took to have someone for the two weeks that were above what they pay you. In the end the damages seem not to be much,, you posted this under California law with a Colorado Zip code, if you need help in CA, I may be able to assist

If you are in California, the answer is that you are an at will employee and can leave whenever you want, unless you have a written contract that says other wise.

No one can be forced to provide personal services, so your employer can not force you to remain at the nanny job (which is a type of personal service), especially after they terminate you. They might be able to stop you from working for someone else or get money damages for actual expenses they incur as a result of your leaving. Under the circumstances you have described, however, it is unlikely that either of those remedies would be applicable.

It would be best that the new employer understands you need to give notice to your previous employer and can not start until the notice period is up. Most will appreciate that and expect that same courtesy from you when you leave them.