What would your policy be had you made this appointment a different day of the week? Would you close but the families still pay? How much notice do you have to give? They same policy would apply to this day. You should give notice of any appointment on the Fridays even if she is not expected to use the day, give notice so she knows that she cannot use the day. If she would have to pay for you to go on a Monday then she would pay for you to go on the Friday. If you typically close with no pay then she would get the day with no pay, even if she wasn't supposed to use the day and it didn't technically impact her.

I would clarify all the above and in the future let her know you booked an appointment on the Friday so she knows the day is not available. if you bring the other kids with you then let her know you planned it for the day she was planning not to use so that you didn't have to close a different day of the week and impact her.