Reading the Rough Guide, I don't see much to visit in Denmark, outside of Copenhagen. That is, Copenhagen with a couple of out of town day trips by the supposedly excellent rail system.
The biggest draw to me would be the Copenhagen Jazz Festival, which in 2014 is July 4-13. That's much later than I had figured on visiting; had been thinking in terms of late-May start, home by early July. But suppose we made Copenhagen the last stop on the trip rather than the first as I'd assumed...?