I would ask that family if they can keep the child home that day, since they have been keeping her home frequently. If they can't then one of the new kids will have to start the next day.

I would also send home a amendment to your contract (at least for this family) noting that with the drastic increase in sick days in the last few months your income has taken a big hit and you can no longer allow free sick days. Children will now have to pay when their child is home sick. Have them sign to the new terms.