Copenhagen has a bunch of nightlife activitites. You can read about some of them here. Accomodation in Copenhagen is really expensive. The hotels are really good but if you're travelling on a tight budget I'd suggest checking out Airbnb or hostels.com. Cheapest way to travel from Billund to Copenhagen would be by bus or train. Just ask for help in the airport. People are really nice.
Bus from Billund to Vejle ~ 7€
Bus from Vejle to Copenhagen ~ 17€ (book well in advance on flixbus.dk)
Train from Vejle to Copenhagen ~ 12€ (book well in advance on dsb.dk, only "orange tickets" (tickets that are booked early) are that cheap)
I'm from Denmark. Grew up in Randboeldal (Randboel Valley) approx. 10 km from Billund. The nature is pretty great in this area (forest valleys and lakes) combined with historical Viking landmarks (Rune Stones, Viking ships, etc). If you visit places even closer to the sea in Eastern Jutland, the nature will be great as well.
I suppose you already know about attractions in Billund like:
Legoland
Lego House
Lalandia
