I personally did not think Copenhagen was particularly expensive, Paris yes, but Copenhagen? No surprises.
We stayed at Hotel SP34.. would recommend, very scandi, good prices and happy hour on an evening. Has a fabulous organic Danish continental breakfast.
Sticks 'n' Sushi is a great restaurant, you take your shoes off and the Sushi is great - they have a great rooftop bar too!
What else..Tivoli gardens is a must, even if you don't do themeparks (I don't but the gardens are beautiful).
Don't waste your time going to see the Mermaid.. she's so tiny and the view behind isn't great, didn't see the point in that one.
Hope that helps a little, enjoy! It's a beautiful place and the people are extremely friendly and so happy.