Is she looking to be a long term family or is she waiting to get her youngest into the same daycare centre her older child is in?

The answer gives you two options. If she is going to be more short term then giving her your old rate now means it will only be a few months before you would be replacing her and could give your higher rate to the new client at that time.

The second option is to insist on the higher amount because she is going to be short term and it helps bank extra money since you will need to interview sooner than expected.

Either way I would be careful about the longevity of the family. You mentioned you interviewed two families. You didn't say if the other family was willing to pay your new rate or not. But before making any decision I would be asking both families more questions about their plans for the future and their expectations of you.