For a place not much warmer or drier than the UK a lot of the best stuff is outside.
We go every year as friends live there. The best wet weather thing we did last year was the Danish Architecture Museum. It’s an interesting building and the exhibitions are engaging – rather than looking at a room of thirty chairs like in the Design Museum you move through spaces showing how Danish architectural design has influenced other design, and why it is how it is.
It spirals up through the building, then at the top you ride a helter skelter back down.
I’d not bother with Malmö. Christiania is an awful place, and outdoors.
Louisiana is great but really benefits from a warm sunny day to get the most out of the setting.
Aamanns do the best Snorrebrod.
Blagardsgade is a good place for restaurants and bars, as is the meatpacking district. Hooked and Kodbyens are good casual seafood places but you’ll probably need to book. If the weather brightens up there are some good street food areas at Refshalhoen (which you’ll need to cycle or take the ferry to – we use Bolt or Tier bikes and it’s very easy) or by Nordatlantens Brygge where there’s an outpost of Hooked.