Hmm, You're staying 'til mid-july right? There are some events coming up before that. I'll try to remember some.
I'm not much into restaurants but one that's really good is "Lo Scudetto". Italian food obviously. You should check the menu before getting there though, some of the food is rather odd.
I think you should pay a visit to "Skansen" it's a sort of a Zoo but it's much more genuine. I really like it. One of the institutions in swedish summer is "Allsång på Skansen", but I doubt that people who haven't seen all the hype would understand it. It's like introducing a rookie to football.
I assume you know "Stadshuset", what few people know is that you can get to the top of it and see the whole town from there. The view is awesome. A must do.
Footballwise, I think you would find it interesting to watch Hammarby-DIF 18 May. Always a great atmosphere, it will be even better this year as it's played on Söderstadion.
I'm not sure if "Fåfängan" has opened yet but it's a great café with a nice view.
Then we have this beautiful village called "Saltsjöbaden". A real must see.
Honestly it is. I could give you a guided tour.
That's all for now.