My first thought is that sleeping while the kids sleep is understandable, but not terribly professional, and the fact that she is doing it during the first week when she should be on the ball trying to impress you does not speak well for her. My second thought is that she is 22, and she is young enough that perhaps talking to her and explaining how unacceptable this is will correct the behavior. my third thought is that, given that you stretched what you planned to spend on a nanny to pay for this young lady, she needs to live up to the rates that she set.
Overall, I would definitely talk to her about this, and about the fact that you paid her x amount because she marketed herself as a professional and educated young woman. She needs to live up to the image that she projected during the interview process. If there is another slip up in terms of professionalism and getting the job done, I would not hesitate to let her go.
--Nanny
I agree with this. I would add that it is a huge red flag that she did this on her third day. Stay open to replacing her if she does this again. You mention twice that she is great with kids, but you have only employed her for three days and she's already failed to adequately do her job. I would not call that great with kids or professional and definitely not worth a higher than market rate.