There's a regular Kindergarten in Braunlage, however you should check if they take guests. I did see some private ads for babysitter services, I'd ask the tourist information for contacts.
If you intend on snowboarding while your child is in a new kindergarten environment and you potentially need to be back fast, I'd suggest Kühtai. The kindergarten there is ski in ski out and you're never more than 15 minutes away...