I love Copenhagen. I also love art and design, and eating well and beer:
I like to visit the Mikkeller bar in the middle of the city. The name of the neighborhood is escaping me at the moment.
I find the street food on Papieronen (sp? - "Paper Island" in English) to be a great place to eat and hang out on a weekend evening. (
Copenhagen Street Food) This may not apply as much in the winter, though, since it may be a bit cold to sit outside (I've been there in the summer and spring, but not winter.) On the way there, you will walk past Noma, where you could ponder what else you could be eating if it were possible to get a reservation.
I also like Warpigs, the BBQ / beer place in the market area near the train station.
The Louisana Museum of Modern Art, a bit north of the city, is well worth a visit if you like modern art. It is an easy train ride (about 30-40 minutes, I think), and it is also an easy bike ride ("National Cycle Route 9". Helsingor, aka Hamlet's castle, is about 15km past it. The two of those would make a nice day.
If I remember more, I'll mention it.