Overpaid in Stamford?

I have a salary question for you. I have a nanny who works from 7-810 AM and 2-7PM, five days per week. It is a split shift. When looking to fill this spot, it seemed impossible. Those who would accept what I thought fair did not seem fit for the job. Those who I thought fitting for the job asked for what I thought too much. I ended up going with the individual I thought was the best for the job. I am paying her $800.00 off the books per week. She takes the train in from Pelham every day. She has only been with us a short while. On days when I need her to work straight through, such as school being cancelled for snow or a school recess, she is demanding to be paid extra. I think that $800 is a fair salary, especially since she has so much free time on her hands. She says that I told her hours would be free during the day and she can work for other families and make money during that time so that this is not fair to her.

Who is right? We live in Stamford, CT. I have two children in school full time.