Babysitters normally charge a higher hourly rate when it is just for a few hours. When you are paying for full time, it is normally a set amount per day or per week.
When my daughter was that age, she was in a private home daycare. I paid $120 per week and provided her food. This provider was licensed and had all the necessary certifications. If I had two kids there would have been a discount.
You gain the convenience of this person coming to you (cuts down on travel time and gas) but you also have the added expense of food for the kids and the babysitter. I would not pay more than what you pay your normal daycare unless absolutely necessary. She is helping you when you need her so that is a bonus but again, you are paying her so she's gaining something too.
I would ask the babysitter what she would like to be paid and go from there. If she says anything less than $350 I would pay her $350 (she would be thrilled). If she says anything over $350, I would pay her what she asks (or a bit more) up to $420 (since that is what you would pay at her daycare). If she asks for over $420, I would play that by ear...If she aks for $450 you could either say"well I was thinking $375, can we meet in the middle at $425" or you could say "I was thinking $350, instead of meeting in the middle, I can go as high as $420, is that ok? Or something along those lines.
Alot of people have commented about nannycare being more expensive but I didn't read "nanny" anywhere in your post so I am assuming you are utilizing someone you know as a babysitter not a professional nanny. Under that assumption, I am figuring this is tax free money to this person AND you can't claim it on your taxes like you can your daycare. If my assumption is wrong and this is a professional nanny, she will tell you how much she charges.