It's worth knowing that most major hotels in Stockholm offer a babysitting service.
A good place to start looking for a babysitter is in the listings of the English language site The Local:
Or on the "family care" website, which allows you to search listings of young people looking for work with kids and families:
The "Stockholm-Qui veut" site is a great French-language resource:
Childcare services are provided by the Swedish state for all children from one year of age. Parents pay a fee which is calculated according to their income. You should contact your local municipality for more information. You can find telephone numbers and addresses in the green section of the Stockholm Yellow Pages.
In addition there are a number of private English-speaking preschools in Stockholm, such as: