You've got a fairly unusual situation, but coming at it from just the numbers, it's a huge rate increase. I think (not in DC area) the usual increase for a new child is $1-2 or about that.
If you're offering $20 an hour you might as well get your own nanny who will come to your house (it sounds like you bring your kid to hers?), do kit related tasks and who won't depend on her own children's schedule, etc.
If you're very happy with her, though, of course, meet her in the middle. She's negotiating, so you negotiate as well.
But the fact that your oldest will be in preschool 9 hrs a week shouldn't be a factor - presumably you would still need her for all the days that the school is out, the kid is sick, snow days, etc, so you would be unhappy if she took another job for those 9 hours. Unless, of course, you plan to cover those days yourselves.