You should either accept this new salary or look for another job. When you find one you can give notice to your current employer.
As a nanny you are basically a babysitter who works in "their" home and not yours. They have every right to tell you that they are not paying you the same way.
I was a nanny for several years. I am still very close to the family. I got paid by the hour, per child in the home for that day but not for school kids over 8 years old. If a child stayed home from school sick I would not be responsible for them. They were pretty much able to watch themselves and didn't need me to tell them what to do. They had their own phone line and could call mom, a professor of nursing at a local college. If I was not there the school kids would have been home alone if they had a fever or something like that. If they were very sick they of course went to the doc with mom or dad.
I also got paid separately for different chores I would do like starting dinner or loads of laundry.
If I had 2 boys all day (8 hrs.) and one girl after kindergarten (3 1/2 hrs) then I got paid by the hour for 3 kids for the amount of time they were in my care.
Then I got paid per load of laundry I washed and per load of clothes I dried and put up. There were 7 kids and 2 adults in the household so there was always loads and loads of laundry to do. I was always honest so she could trust me.
Then I got paid for starting dinner, only paid for 1 hour. I often only charged 1/2 hour though.. Mom worked 8-5 and dad worked from 7-3. He was always home right after school, by 3:30pm. He was often very glad that the prep work was done and meat was cooking. If he didn't want me to start dinner then I did not get paid for that chore.
The kids did all the cleaning when they got home from school with dad supervising so I didn't have any cleaning to do. If I did that she would have paid me more. I did do some office work for mom when the kids were napping but that was not every week or even every month.
I was free to take the kids wherever I wanted, we could go on field trips, go shopping, out to eat, I often took the kids to rehearsal for the group I sang in, they also went to every performance. I was their surrogate mom, not just a staff person.
When the kids all stated kindergarten I only worked half a day and they had him in mornings so that I could come in at noon. I still only got paid for the jobs I performed.