Adam,
I would agree with Graham about the Viking Ship Museeum in Roskilde (
http://www.vikingeskibsmuseet.dk/), and at the same time make a tour of the Roskilde Cathedral where many all Danish kings and queens are buried (
http://www.roskildedomkirke.dk/dk/).
Regarding WW2/military/aircraft interests, I would pass the Museum of Danish Resistance 1940-45 (
http://www.nationalmuseet.dk/sw10155.asp) and the Technical Museum, Aircraft Collection in Elsinore (
http://www.tekniskmuseum.dk/, nothing much in english: chose "Samlingerne" and "Flysamlingen..")
If Tollund-manden (
http://www.silkeborgmuseum.dk/), not exactly near Copenhagen) is of interest, I would also pass the National Museeum (
www.nationalmuseet.dk). And if in Jutland to see Tollund manden, why not pass Aarhus and Grauballe manden (
http://moesgaard.hum.au.dk/my.php?top=13&language=1).
That was just a few suggestions, which did not include: Kronborg in Elsinore, the royal palaces and many other odd places.
Regards,
Mikkel Plannthin, Copenhagen