Have her come into work and do child related (if you had a contract all that would be spelled out in the contract) duties during her regular work hours during the week

But yes, pay her.

This mom reminds me I used to work for a family that refused to pay when the kids nap. Two hours nap per day most days, the mom will shorten my check by ten hours and one day, I just didn't show up.

Expectations should have been made clear in the interview process. Since they were not, of course you have to pay her.

And this is another thing. If you decide to travel for your business trip and would decide to leave your kids with your nanny; you should ask her if she is willing to stay overnights and working 24/7 for the days you will be away. Same thing. When you hired her, you should been very clear about it as well. Asking the nanny is she will take a position when she would needed stay overnight when you would needed it; and telling her that your work would requiere traveling and you would need someone who is able to stay overnights as I said everytime when you need to travel.