I intend to fly down to Copenhagen for a day from London and plan to spend about 10 or so hours of sightseeing.
Since this will be my first time there and the fact that I want to reach as many of the top tourist sites in Copenhagen during that time, please which places will you suggest that I visit?
Tivoli Gardens in the evening - it's very atmospheric at dusk and at night. Not so magical on a rainy morning.
A boat trip on the canals gives a good look at the city.
Then I'd hoof it around on foot and try to see the sights - head NorthEast from the station / tivoli and you can see Nyhavn (pretty street split by a canal), Amalienborg Palace, the Little Mermaid. Just here there's also a star castle fortress if that's your sort of thing. Then head back SouthWest to Roseborg Castle, the Round Tower (interesting story about Catherine the Great), and back to city hall and the Hans Christian Andersen statue and you're back at Tivoli.
If you've time, you could add Vor Frelsers Kirke (our saviour's church) over the water to the East from the centre, still walkable. It has a corkcrew spire with a staircase on the outside, a good viewpoint for the city.