Kungsholmen is a very developed island and for a runner that means traffic lights and stops. For an uninterrupted run, the easiest selection is the about 10k trip around the island. The route has a flat section but some hillier sections in Fredhäll.
For a shorter and almost flat route, you can modify the route and skip the hills of Fredhäll.
For inspiration you can have a look at the route of the annual Kungsholmen runt running event. That event is worth participating if you happen to be around in Stockholm in early May. It’s a fast course. A very nice way to test your speed after the winter!
Mynksteri's site 