Well, you're charging them more than a child care center would. They'd pay by the week no matter what though. Even if they miss a day they have to pay for their child to have that spot.
You are not a babysitter if you're going to their home, you're a nanny and those jobs include light housekeeping every single day. It's your job to cook and wash the dishes you use but I wouldn't think it's your job to clean their house overall. That's a totally different job description. That would pay you more for sure, you'd be getting paid for house keeping duties plus the kid duties. So stop, just don't do more than you dirty. If you're loading the dishwasher and there is room for a few more pieces then it would seem petty to not pop the other stuff in there.
When I worked as a nanny I was contract labor. They paid me X amount of dollars for just watching the kids and then I had a list of my extra duties. I got paid a specific amount for each load of laundry I did and a different fee for each day that the mom had me start their dinner. I took the kids with me everywhere I wanted/needed to go. I didn't charge the family gas money because I was going places that "I" was going.
They didn't hold out taxes on me but we did do the paperwork for them to file the child care benefit on their taxes. I also claimed every dime I made.
In child care most kids are there 10.5 hours too. They usually go in before mom/dad's work so around 7:15am/7:30am then don't get picked up until mom/dad get off work then fight traffic. So hardly ever picked up before 5:30pm/5:45pm. Plus they get 2 full meals plus 2-3 snacks. They have fully trained and certified teachers who have education before they're ever hired for their jobs and have done 20+hours of continuing education each year to keep their jobs. They also follow a purchased curriculum that will prepare the 5 year old for school, again, if he's not already a full time student each day for either Pre-K or in Kindergarten.
And they get minimum wage or just above it. A toddler room worker has 6 toddlers, 2 year old classroom has 8 kids per teacher, the 3 year old teachers have 12 kids each, and still gets only the hourly around minimum wage. A 4 year classroom teacher has 15 kids, if he's in the 5 year old classroom (not in school) then they might have 18 kids per teacher. If he's in kindergarten then when he's not in school they can have up to 20 kids each, all for a good old minimum wage or just above.
So no, I think you are way over charging this family for just child care/picking up after the kids. Yes, I have a bias in favor to a fully licensed child care center. It's actual classrooms and the kids come in ready to learn. They are exposed to much more organized education and they do better in school overall when they've been to child care or at least a preschool style center or Pre-K before Kindergarten.
I think you need to be honest with the family and tell them that you don't want to watch the kids. It's too much for you. Then they'll be free to enroll the kids in full time child care and you won't be having to do so much every day.
You can always call around to the local towns and see what the cost of regular child care would be. Again, they offer full preschool types of classes and have basic first aid/CPR plus certifications/degrees in early childhood development/hours of continuing education hours/and more. Plus they have a lot more kids in their care. T
hen charge them less than they'd pay in regular child care BUT negotiate a contract with them where they pay you every Monday morning, the full weeks pay. Regardless of them staying home during the upcoming week, they should pay you by the week. Then if they don't have the money on hand and in your hand you tell them sorry, and you leave.
Tell them they can take it or leave it.
I think you're trying to do a good job for them and they're trying to make it on what they have.
If you want to check in to it for just general information you can google child care assistance guidelines. It should have a link somewhere to show how much income gets how much assistance. They'll have to use a child care provider the state agrees to pay though. But you'd be free to not have a nanny job.