Having spent over a year in Copenhagen, I think it's well worth spending four days there. Gabelle pointed out the best attractions that you will find. I think the nicest part of Copenhagen is walking around Stroget (and surrounding streets) and just living like a Dane. Spend time sitting in the cafes and people watching. Stop in one of their candy stores...Danes love sweets. Visit the extraordinary museums in the city. Walk down through Nyhavn and grab a beer with the locals. Stroll through Tivoli and be sure to find an aebleskiver while you're there. The Danish people are very welcoming, especially if you take an interest in their culture. They may be the people who have the strongest national pride that I have ever had the pleasure of meeting. Enjoy the city and please ask if you have any questions!