I used to live in Copenhagen and loved it. The people are so friendly, the food is great and the city gorgeous. The Danes (and other Scandies) are super child friendly and we are thinking of taking our LO there this summer for a short break.
Bear in mind that DK is expensive. What do you fancy doing? Would you like to stay in the city? Or rather on a beach?
Try: http://www.dancenter.dk/ to get an idea. Sry it's Danish, I can't get the english site up. Uge= week, husdyr= pets, antal sterner= stars, afstand= distance, kyst= beach, indkoeb= shopping. Use http://www.oanda.com/convert/classic to check prices.
Personally, I loved Roskilde. It's the former Danish capital with a beautiful sound, lovely houses, very quaint scandinavian design, great little shops and only a short train ride (40mns) or so from Copenhagen. It also features the cathedral with all the kings' tombs and a great viking museum.