OP - there's not a going rate, this isn't a normal thing for a boss to ask.
There's your answer, you are actually in a strong position. They want the flexibility of a nanny, they want someone who'll come into their home, work longer hours than a childminder, offer the dedicated care of a nanny, they want someone who'll cover holidays and sick days, like a nanny. They just don't want to pay for it. If you say no, you expect to be paid for 3 full days even if the youngest is at school.
They are unlikely to make you redundant without checking they can replace you with someone who will do the hours they want, with the flexibility they want.
Just say no, point out this is not a normal arrangement and they are unlikely to get a nanny who'll agree to it. there's a good chance nothing will come of it.
That said, you do need to think about your own career, while a nanny is the best for them now, once the youngest is at junior school, an au pair might be cheaper and more suitable for them. Over the next year or so, speak to the agencies and ask ot be considered for any other roles that come up that suit you.
A nanny doesn't stay with the same employer for 20 years, it's always a limited job.