Yes, I am danish. If you order in good time, you can get a cheap flight (by danish standards at least) from Billund to Copenhagen. Otherwise you would take the bus to Vejle and change to train to Copenhagen.
Since I've never lived in Copenhagen, I am not too familiar with the night life. But of tourist stuff I would recommend Nyhavn, Tivoli, Frederiksborg Castle and Kronborg (Though both are quite a bit from Copenhagen).
The Little Mermaid I guess is mandatory, though many get disappointed, it really is 'little'.







