I lived in Denmark for 4 months and my parents came to visit. Lemme tell you, cheap anything is just not an option. Especially with current exchange rates and the crazy tax rates.
Also, Scandinavian hostels are not your average hostel. They cost more, they are nicer, and 9/10 times they are pretty boring or filled with school groups. Plan on it being JUST a bed for a night...a $30 bed for the night.
Most hostels are part of the Danhostel chain. Copenhagen City hostel is your best bet. It's a high rise right smack dab in the center of town with decent transport connections and within walking distance of Tivoli, the main square, and the shopping street.
If you want a more personalized stay, then do a bed and breakfast.
http://www.copenhagenbedandbreakfast.dk/CopenhagenKortGB.htm My parents did this. It was their first time out of the country. Hostels scare the crap outta them and hotels were way out of their price range. "Bed and breakfasts" are often a bed with no breakfast in an extra room that a person rents out for extra income for around 60-80 euro a night (compare to hotels at 200). My parents loved their room and really enjoyed talking to the owner. It can be a bit hard to meet Danish people as they often keep to themselves and this is a great way to meet some of the locals. Just don't expect maid service and breakfast.