We live in Enskededalen witch is as close as you can get the city centre staying in a house with garden. It is an area of cultural and architectural interest. The wooden houses in the area were build in the 1920s as an opportonity for families to build their own house with a garden near Stockholm. We have bikes if you want to explore the city, look for a beach or just go for a hike. There is a playground and a small pool for the youngest in the park opposite our house and a public pool for the teenagers just one kilometer away. There is also a couple of lakes in the woods a few kilometers from the house.The house has three floors with a lot of light and a nice patio. The garden has a variety of fruit, berries, herbs and flowers (help yourself!). There is an extra room in the basement with bed, there is room enough to invite friends to come over. The area is quiet and charming, with very little traffic and still close to the city. Only 8 minutes walk to the subway and then 10 min to busy city life.
Stockholm is a great city to visit wth a lot of things to do for kids and adults. 
Skogskyrkogården, the woodland cemetry placed on the Unesco World heritage list is just around the corner.
http://www.visitstockholm.com
www.skogskyrkogarden.se/en/
http://www.visitskargarden.se/en/
There is one "must" if you want to see the archipelago and that is a daytrip to Sandhamn with a steamer where you can enjoy the tranquility of the sea and the busy summerlife on the islands.http://www.waxholmsbolaget.se/visitor/archipelago-traffic/plan-trip/
Here´s some more pics of hour house! http://www.flickr.com/photos/nilsberno/
You and your kids are welcome!