Well a feat of engineering to see is the bridge/tunnel over to Malmö
Going out in København is a lot of fun as well. There are plenty of really active pubs to have a lot of fun drinking.
Checkout Christiana as well. Plenty of info about it here:
If you want to see the Little Mermaid statue it is a little small and disappointing but still something to do.
The Carlsberg brewery is cool as well. almost 9 years ago
Consider the tivoli gardens (http://www.tivoli.dk/composite-3351.htm) and a lunch at Ida Davidsen (http://www.idadavidsen.dk/en-gb/cms/Welcome) - both things dont come cheap, but are a experience to be tried ;-)
There is also "Bakken" (http://www.bakken.dk/english), Lousianna (http://www.louisiana.dk/dk/Service+Menu+Right/English) both of these being out of Copenhagen, but within reach of public transportation (approx 1 hour) which generaly are safe and well organized.
In downtown Copenhagen there are a lot of historic buildings and museums - the museums have free admission on wednedays.
Enjoy and have fun.
Remember to try the canaltours of copenhagen - really worthwhile - there are several vendors, two of these being:
b) http://www.stromma.dk/en/CANALTOURS/ over 5 years ago