Copenhagen doesn't shout. It is a capital of about 1.4 million people that most of the world knows from a painted canal and a small bronze mermaid, and it turns out to be much more interesting than either. Bishop Absalon built a castle on a little island here in 1167, and the city has spent the 850-odd years since being burned, besieged, rebuilt and tidied. Christian IV, the king with a taste for red brick and towers, gave it most of the skyline you see in the postcards, and two big palace fires, in 1794 and 1884, gave it the rest.
The character is a mix of royal stage set and very ordinary life. Kings still live in Amalienborg, the Parliament sits in the same palace as the Supreme Court and the Prime Minister's office, and a few streets away people park their bicycles outside a bakery that says it is the oldest in town. Five days is the right length because the sights are close together but the city rewards slowing down: a long lunch, a market, a cemetery, a harbour walk. It is also an expensive city, so five days lets you spend where it counts.
This guide is written for November, when Copenhagen is at its coziest and its shortest. The sun rises around 7:30 and sets around 4pm, so every day puts the outdoor sights in the morning and the indoor ones and the lights after dark. Day one is the old center, Rosenborg and Nyhavn. Day two is Christiansborg, Christianshavn and Christiania. Day three is the northern harbour and Tivoli in its Christmas dress. Day four is a trip up the coast to Kronborg and the Louisiana museum. Day five is Torvehallerne, Nørrebro and Vesterbro on two wheels. Roskilde and Malmö are in the extras at the end. Start on a Tuesday: Rosenborg, Kronborg and Louisiana are closed on Mondays, and the Round Tower stays open until 9pm on Tuesday nights. For the wider picture, see Denmark, and for the same winter mood in Sweden, Norway and Germany read 5 Days in Stockholm, 5 Days in Oslo and 5 Days in Hamburg.