Ah! I was in copehnagen earlier this year for a few days. I stayed at 71 Nyhavn - very interesting 200 year old warehouse on the docks converted into a hotel. Fun location. The entire "dock" - nyhavn is a row of restaurants, bars, etc. Check it out.
There is also a main shopping street - but its just the same stuff anywhere else in the world.
We saw a ballet at the opera house - excellent. We were able to buy earlier that afternoon tickets in the 4 or 5th row near the middle. Fantastic!!
There is a large museum there (name eludes me at the moment) with a nice selection of modern art to medieval art.
The little mermaid wasn't as exciting as I had hoped - and its a serious trek (well, i assume you could just take a cab)
We also took a boat tour - starts right at nyhavn and is worthwhile.