When I get the measles event (or even worse, the poison event!), I don't bother trying to find a doctor to keep him at work. I just keep picking up kids and dropping them in the crowd of sick kids.

Sometimes it doesn't work, but often it does, and then I have a cured kids plus another with some healing skills.

I just keep doing that until all the kids are cured (I may use the detailed screen to find which ones are still sick if needed.)

Eventually, adults with healing skills (whether checked or not) will come to the "clinic" and help out.