We have from morning until about 5:00 pm in Copenhagen. What are the must sees?
So little time!
I would highly recommend a canal tour to see many of the main sights of the city. Departs from either Gammel Strand or Nyhavn.
Spend some time walking around Nyhavn.
Find a traditional danish restaurant serving smorrebrod sandwiches (and if you want to go local herrings and Danish acquavit). A favorite is Restaurant Schonnemanns.
Don't waste time at the Little Mermaid.
Resolve to come back and spend more time!
Are you by any chance on a cruise? If so, be sure to get Rick Steves Northern European Cruise Ports. He has all the tips you will need for a short visit, including prioritizing sights and getting between them.
If you're not on a cruise, or just want a preview of the book, here's his Copenhagen page; click "At A Glance" for his recommended highlights: http://www.ricksteves.com/europe/denmark/copenhagen