While she may have breached the contract by announcing her decision to quit w/o notice after failing to come as required, you have no documentation of that. While an e-mail generates a message, it does not have the proof of receipt that would best help you.
If you send your ex-nanny a letter detailing her failure to appear when required; your call to her to ascertain the reason for that failure; her announcement in that call that she had quit and that said announcement did not meet the 3-week notice in the contract, you will have set forth the elements of the breach of contract.
That should be sent by Certified Mail / Return Receipt Requested and First Class Mail, using the proof of mailing service the USPS offers. While this will strike you as redundant, note that people will try to duck Certified Mail if they are expecting to be told something they don't want to hear.
The second letter provides proof that a letter was sent to a specific address on a specific date. As a letter sent is presumed received absent evidence to the contrary, such as it being returned, you would have a back-up proof of service to the green card from the Cert. Mail.
The object is, of course, to establish the facts before she can fabricate an excuse for accusing you of breaching the contract.
You may wish to review your situation with an employment law attorney. Your state and local bar associations can assist you.