I would get out on a boat and go and cruise up to see Birka , lovely trip and a great chance to see the remains of a Viking town.
But even ahead of that is Djurgarden , Skansen is amazing the buildings, the animals the pictures of Swedish life, the Vasa is one of the best historic ships you will ever see and it is a beautiful island to see, lovely walks.
Gamla stan is a real rarity in Western Europe a perfectly preserved centre rare in capitals all touched by war, great bars, royal palace and places just to wander.
I lived there for 6 months , I loved it.
I also liked wandering around sodermalm and just walking all around the city .
There are also great museums, go and see some sport, there is the hall where the Nobel prizes are awarded and various little nick nacks everywhere