Let's face it: You can't visit the Danish capital without seeing the Little Mermaid, so head along the waterfront from Nyhavn to Kastellet (less than two kilometers) and take in the iconic statue and surroundings.
Kastellet is the former Citadel of Frederikshavn, the oldest parts of which date from 1625. The Citadel buildings are well maintained and well worth exploring. The Little Mermaid (Den lille Havfrue), which you'll see from the shore, is the official emblem of Copenhagen.