We were there for a week a couple of years ago and loved it. It's a very 'walkable' city. We don't do late night drinking/clubs etc - enjoy pootling and eating, but would agree with Tivoli - the setup is lovely and entry is included on the Copenhagen card so if you had a card you could go by day and for a drink after dark.
Rosenborg Castle is stunning and if it's a nice day the park it's in is nice for a walk/coffee.
The food market is great for lunch (the hotdogs are amazing, even from street vendors. Not Michelin starred but fab for lunch on the hoof).
The shopping is good - there are lots of small boutique shops dotted everywhere, good design, clothes (I went mad in Masai's discount store).
You may not have time, but one of the highlights for me was a trip to Isak Dinesen's house - it's just outside Copenhagen, about 20 mins on the train, but if I could choose a house to live and die in, that would be it.
We had a delicious smorgasbord lunch one day near Nyrhavn, but of course I can't remember the name of the café! It may have been Café Orstrup on Holbergsgade.
Café culture is amazing - there are hundreds of brilliant coffee shops and bakeries - I ate my own substantial bodyweight in rhubarb pastries.
Nyrhavn is good for people watching and a lot of the outdoor terraces there have blankets and space heaters so you should be OK even in February.
Hope you have a lovely time.