If you have more than a few hours, try to get out of then inner city. I find that e.g. Strøget (the shopping mall) is much like any other shopping street in other European city.
Many find Christiania is fun to see. It's a special "free" part of the city, where people, mostly hippies and activists live in "alternative" ways.
Go to Nørrebro (where I live). Cosy Cafes and a chance to see how people live. 10 min walk from the city, next to the lakes.
Personally I don't think it's worth to see the little mermaid. Not if you are short of time. If you take the boat ride, do take the longer, which will take you by the mermaide. Then you can tell your friends you saw it :-)
If you are in to museums (and the weather is bad) go see Nationalmuseet (the National Museum) for a bit on Danish history and culture.
I'm told the dance/party scene is good. But I'm too old for that.
For jazz concerts go Copenhagen Jazz House.
Bring a rain coat and umbrella.