I think you should approach it as if this were a continueing relationship. After all, you never know what the future holds. I always pay for state holidays, since I get paid as well. For sick days, we (nanny and I) agreed that she could take up to 10 days a year as paid sick time, so maybe you should offer 1 per month? For days when child was sick, my nanny usually got extra pay if they were available for extra hours so I could still go to work. one of the advantages of having a nanny is that they can still care for a sick kid. When I was sick and stayed home I would still have nanny come so I could take of myself without having to worry about child. Given the short-term nature of the contract, I wouldn't feel it necessary to offer paid vacation time. For shortened hours, I always paid if it were my "fault" that they were shorter -- e.g. I chose to take child to the doctor instead of sending the nanny. If the nanny needed time off for a doctor's visit, then generally I didn't pay for that time.
My overall working theory was that if it wasn't the nanny's "fault" the time was different, then her pay shouldn't suffer. That just seemed fair to me.