I gave it a try. I modified my payroll spreadsheet. If this appeals to you and your wife, I'd suggest buying time cards (or printing some), and having one for each child. Then transfer the daily info to the spreadsheet. I tried to do a minimal amount of work just to throw it back at you to see if it was at all appealing. I could give you some ideas/tips on how to quickly add more kids, or how to add sheet for the bi-weekly ones. There would be two sheets for each child, one being a pair of stubs - one for the client, one for you to file away. This is what I do for my employees. I also have a sheet that totals up all the info for everyone, you could do that if you wish.

Edit: Couple more comments.
You have to enter info in the Time Input table, the hours shown on the stubs show the values entered in the time input table. If you don't enter data there to match up with the current billing period, the stubs won;t show the correct info.
I didn't hide the data validation list (Yes/No) just so you could see where they were, a person could change the font color to white.
Play with the hours in the main table, if you enter less than the minimum hours, it will calculate the total based off the minimum hours times the rate entered. If you change Full time to No, it will calculate the total simply by multiplying the hours times the rate.

I like your use of a pivot to summarize the data, I just couldn't figure out a way to deal with the minimum hours with that method. Then I thought I could modify my payroll spreadsheet to accomplish the result

Daycare_V2.xlsx (13KB - 200 downloads)