Tivoli would normally be closed at this time of year, but as it's a big anniversary year, they are opening in February. It opens 2nd Feb.
http://www.tivoli.dk/ It's a nice place to walk around but costs about a tenner to get in, without the ticket for the amusement rides. Some of them are really good though!
Take a boat trip to see a fair amount of the city easily and quickly. If you take the blue Netto boats, they cost less than half the price of the DFDS ones, and they depart from just near eachother on Nyhavn. Nyhavn, the picturesque little harbour is worth taking a walk along. The bars are really expensive there though not surprisingly, so save the beers and food for elsewhere in the city.
The little mermaid can be seen from the boat trips, so avoid the long walk there in the cold at this time of year. It's pretty disappointing anyway! If you do walk out there, walk from Nyhavn, through the royal palace Amalienborg, and then on through Kastellet, to the mermaid. Kastellet is really nice, an old fort with ramparts, with some great views of the harbour.
Papirøen street food is closed for good now, shut down at the end of the autumn. Will be re-developed, no doubt into stupidly expensive apartments the city doesn't need!
Christiania can be pleasant, and at the very least it's an interesting place to have a walk around. Pusher street where the weed is sold generally isn't though, as it's just gangsters being typical gangster dicks.
Copenhagen isn't the best though at this time of year, due to the dark and cold. Summer is when to visit, as the place just transforms into a massive party zone.
Avoid the Irish pubs on the walking street, the pubs on the strip between Town Hall square and central station, the tourist traps on Nyhavn, and you'll be able to find pints similar to London prices, and if you don't mind smoking bars, there's tons of cheap places too (the smoking ban doesn't apply to half the places here!)