For indoor activivtees, on a rainy day (and they come often 😉 ) there are a number of really good museums.
The National Museum is right next to the parlament in Christiansborg. They have collections with Danish history (of course), but also a lot of stuff from Greenland, a long with things from all over the world. There are a good section, where kids can play, and of course some Viking things. The museum is a bit dusty, but have a nice shop.
Copenhagen Museum is closed, and will first open up in 2019. They have a web page, where you can find fantastic old pictures. We are looking forward to see it again.
If you are in the center at Nyhavn, you can visit the old lightship Gedser Rev. It was placed in the sea, warning passing ships of the reef at the southern most point in Denmark at Gedser. https://en.natmus.dk/museums/the-ships/gedser-rev/
Gold and diamonds are everybodys best friend. You can find the crown jewels at Rosenborg castle, near the Kings garden. There is also a collection of old royal stuff: the old paintings, the Thrones, furniture and other stuff. So if you are into the european royal familys, this is a must. http://www.kongernessamling.dk/en/rosenborg/visit-rosenborg/
Outside of Copenhagen
Outside of Copenhagen there are three places worth visiting. All are within reach of one hour by train.
The Kronborg Castle in Elsinore (Helsingør), is where Shakespeares Hamlet takes place. There are great views over the Øresund to Sweden, and there is also the Maritime Museum next to it. Two good reasons to go to Elsinore. Train to Helsingor
In Roskilde you can find the Viking ship museum, with the restorede ships, found in the Fjord. A lot of information about the vikings sailing out and trading and plundering. In roskilde there is also the Roskilde Cathedral, where the tombs are for all danish Queens and Kings, for almost a thousend years. Here you can really feel the wings of history. Train towards Roskilde
In Hillerød, north of Copenhagen, there is the Frederiksborg Castle, The National Historical Museum, that also have modern art, and a barok garden. The park and the Castle itself are really spectaluar, by the lake. S-train to Hillerød.