At minimum it is two times the daily rate plus. You are only open 9 1/2 hours anyways so they are coming pretty close to three times your daily rate. Taking into account the child will be asleep some of the time with you $100 a shift is not unreasonable. I would also consider some sort of premium for the weekend and any stat day since they will likely be getting a shift premium too on those days.

Do realize that it will be taking up a full time space so you might also consider doing that. In essence they are working their shifts and getting what they consider full time wages on a monthly basis. They will be taking up a full time space. Charging them your regular monthly rate is an option with the understanding the weekends and stats are included in that rate and set out how much extra time they may use if desired such as bringing the child for an extra day when they have errands and it not costing them since they are paying for the space anyways.

If you will have the yearly calendar then count up how many days you will be doing care and then since it is 24 hours double that to determine the monthly rate. Or do that for the days that fall during the week and then apply a different rate to days that fall on weekends or stats.

It isn't something I would want to do. I have had kids for evening shifts - 4-10 but they were for desparate neighbours etc and people that were already friends and they paid well for the privilege.