I provide extended overnights for the foster system, and they pay me $100 a day for older children. My friend provides overnight foster care for infants and toddlers, she gets $250 a day per child. I also have private clients that I charge $100 a day for overnights, but not for a full 24 hour period. I have a private caregiver that in emergencies will take my child overnight, and I pay her $100 to take her from 5pm to 7am, so it is not even a full 24hours.
While it may seem silly to pay someone for sleeping, that is not really the case. You are paying that caregiver to be on call to wake up and take care of any emergency or non-emergency that might happen. I definitely do not sleep as heavily when I am responsible for someone else's child. In fact, the first few times I did it, I could not sleep at all. Also, you are paying her to stay put at your house. If she were not providing overnight care, she would have the freedom to come and go as she wished in her non-working hours. But because she is obligated to stay at your home for the safety of your child, she is technically working even while sleeping. So I think to say that she would be just be sleeping anyways is a tad disrespectful about the value of her time.
I have another friend who works in a residential program with elementary age youths, and works four 24 hour shifts in a row. During the day, she receives her normal salary, which is about 18.00 an hour. At night, for the time she is sleeping, she makes half her wage, or at least minimum wage.(whichever happens to be higher) If she has to wake up for any reason at all to provide care, that time awake is logged at her full salary. Perhaps you could do something like that. As good as your caregiver is, a two year old may have extreme anxiety after the first night if she has not experienced the pattern of an extended trip, and may not be able to be soothed to sleep the next night.
Your caregiver deserves to be paid as any professional would if she is as good as you portray.
Hope this helps and hope you have a great getaway!