The correct pronounciation is Copenhaygen . It is still a mystery to us why Danny Kaye sang "Wonderful Wonderful Copenhahgen". Maybe he was trying to sound Danish, who knows. The song was from the movie "Hans Christian Andersen" and 2005 will be the 200th anniversary of Andersens birth so come back and see us and join the celebration. We'll understand what you're saying even if you call our capital Copenhahgen.
We pronunce it Co-penhown and it, of course, has nothing to do with us being lazy and everything to do with it being spelled København in Danish.