I will recommend Kronborg Castle (Helsingør) in North or Møns Klint in South for an daytrip from Copenhagen.
Here is a list of 25 recommendations to do in Denmark:
Happy holiday!. about 5 years ago
Attitude and personality - Copenhagen on a shoestring
http://izi.travel/browse/d81af9ce-d64a-466a-8f7b-919b9210c94a?lang=en over 4 years ago
I will also recommend visiting Roskilde Cathedral which is a world heritage site.
Best almost 5 years agoAnswered by Danni Jensen via WorldNomads.com