OP, what you may want to look for is a "Nanny Manager" who can provide both care for your children when needed and who can capably offer household management services (laundry, errands, grocery shopping, being around for repairmen, whatever else you, as a single parent need a "spouse" to manage) while the kids are in school.
If you have the space to house a nanny manager, make the position live in, and give her her "on call" schedule as far in advance as possible. Yes, you will need to pay her to be available during those hours, but if you are already paying a reasonable hourly rate and have chosen to guarantee her 30 - 40 hours a week (Paid even when not needed), she isn't likely to demand OT or even straight time for the on-call work. She might be willing to take half-time pay to just be available.
It's not impossible, it can be done, you just need to find the right person to do the jobs you need done. And it doesn't have to cost you six figures, either!