I agree with everyone here. If she is paying for that day then you need to be available for her. If she needs to come and you cannot take her child then you need to reimburse her for the day.

When I book a day off I let ALL parents know. If you booked a day off for an appointment then even if she didn't need care that day she shouldn't be paying you. What about any other kids you have? Were they all charged for the day as well?