I am a licensed historical guide in Copenhagen. Copenhagen is an old town with a lot of history. And I do like to tell about city's development over 1000 years. We have beautiful castles from 1600s in the center of the city, as well as palaces from the 1700s. and other . Good expamles of Renaissance and Baroque. Other interesting sights could be The little Mermaid and the Gefion Fontaine.
If you like to go out of the city, I can suggest a tour to Kronborg, known from the play Hamlet. It is located 40 km north of Copenhagen.