In reply to MHutch:
First off watch out for this lot www.bbc.co.uk/cbbc/horriblehistories/song16.shtml
Secondly , we did this sort of thing two years ago, ferry to Jutland then drive to Copenhagen before going on to southern Sweden. Staying with family rather than camping (kids were a little young then). We're planning on doing it again next year but camping so let us know how that goes!
Roads are easy and quiet, rest stop/services are lovely and the food is great, really puts the UK to shame. Bridges Jutland to Copenhagen and CPH to Malmo are so quick but yes a little pricy.
The Viking ship museum at Roskilde is a must, dressing up, archery (son got stung by a bee but hey ho!) Swarthy blokes wielding axes (wife liked that bit). Just amazing to see how small the viking boats were and how far they got.
Aarhus has an old town worth a look
Never been to Legoland but Billund is a nice enough town (Jutland is really very flat so good for cycling!)
In Copenhagen
Tivoli is a must! rollercoasters to stalls to food.
Nyhavn - sitting in the old dock drinking a cold Tuborg... (best bet nip up a side street to buy your beer as the prices on the dockside cafes are outrageous!)
Carlsberg brewery in Copenhagen (fizzy drinks for the children!)
The boats round the docks, past the little mermaid, children liked them.
Southern Sweden
You really can get into the middle of nowhere very quickly and if you want a few days peace then camping up there would be cool , take really good midge repellent or hope for high winds!