The little mermaid is gorssly overrated, it is small and not that interesting, but Japanese tourists still think it is wonderfull. this homepage has most of the info you probably need: http://www.visitcopenhagen.com/
If you are into opera or architecture, we just got a new operahouse, it is rather spectaculare i think: http://www.operaen.dk/ you can take a small ferry, as it is on the other side of the chanel.
If you visit Strøget, turn left as you reach Kongens Nytorv (the end of Strøget if you started your walk at city hall) then there is a rather large B&O shop on a corner, probably the best one in Copenhagen. When you walk down Strøget, remember to try the smaller streets paralell to it, that is where all the interesting shops and resturants are located, especially around Notre Dame church (Vor Frue Kirke) and the old university main building, which is open to visitors. The shops on Strøget are pretty mainstreem and could be located in any bit city.
Have a nice trip and do not hesitate to write if you got any questions
br Chr.