For the benefit of others reading the thread, rather than just the OP's suggested itinerary, here are some more of our Denmark essentials:
Copenhagen Area - has been covered upthread, but museums, harbour boats, Tivoli, Bakken, beaches, art at Louisiana, Experimentarium, "Fields" for shopping (Strøget is in danger of becoming a centre for tourist tat)... Plus Malmø or Lund in Sweden. And Roskilde for the Viking longboats. See the Visit Copenhagen website for more ideas and events. Bornholm in the Baltic (very sunny island well worth a visit for a few days or a week).
Billund area - All things Lego at Legoland and the brilliant Lego House. WOW Park. Givskud Zoo and Dinosaur Park. Jelling for Vikings. Silkeborg for AQUA, Tollund Man and the lake district.
Day trips from Billund (or alternative bases for Jutland):
Silkeborg (as above). East towards Vejle (Økolariet environmental centre) and beaches, Kolding (Koldinghus Castle and Kolding Storcentre for shopping), Aarhus (the Old Town, Tivoli Friheden, AROS, beaches and more), Randers (Randers Regnskov - tropical biodomes).
Djursland - 2 hours from Billund, but with an airport (Aarhus Airport) that has regular, cheap flights from the UK:
Ebeltoft (charming old fishing town with a harbour and old sailing ships, beaches, glass museum), Djurs Sommerland amusement park (at least as good as Legoland, but obviously not Lego-themed), Grenaa (beaches, great Aquarium).
Jutland West Coast and Southern Jutland (all within 90 mins of Billund):
Miles of sandy beaches stretching north from Esbjerg (e.g. Henne Strand, Vejers Strand). Reasonably-priced summerhouse accommodation in the sand dunes.
Esbjerg (Fishery and Maritime museum is good). Tirpitz WW2 bunker. Several lighthouses that can be climbed. Tønder and Ribe (old Danish towns with cobbled streets). Rømø and Fanø (unique islands that have their own pace and atmosphere, and great beaches).
And then there are the southern islands, Fyn/Odense, the rest of Sjaelland, Aalborg/Skagen in the north of Jutland .... Denmark is much more than just Copenhagen and Legoland!