We have just returned from a 5 day trip. Not my first time.
I agree with others. Copenhagen proper has a lot to offer and the day trips mentioned are easy and quick. Lay out your options and you can decide day by day based on weather, tiredness, spur of the moment interest, etc.
If you want something a bit farther, around 90 minutes by train, Odense is a charming town with Hans Christian Andersen heritage. Not any more of a must-see than the other options mentioned, but I hadn't realized until this trip that it is an option as a day trip.
We were very taken by the Danish Resistance Museum if WWII and its impacts is an interest to you. I mention that as it wasn't quite as obvious in some of the travel literature I reviewed as Copenhagen's art museums, castles, canal tours, etc.
We stayed 2 extra nights when SAS cancelled our return trip to Chicago. It worked out for us with some inconvenience and added expense (I have submitted forms for reimbursement, we will see....). I mention this as SAS has been cancelling on short notice with some regularity. More last summer I understand, but they are apparently still short of planes for any contingencies.