It's a while since I was in Copenhagen, so I can't recommend anything directly, but don't expect anywhere to be really cheap.
A bit of Googling though shows that 'Tight' might be worth considering. There's a limited, but tempting, fixed price menu at 225 Kr (about £27) for 2 courses, or 275 Kr (about £33) for three courses.
Menu (in English) here:
http://tight-cph.dk/files/MenuEnglish2011.pdf
Home page here:
http://tight-cph.dk/
Rated 3rd best restaurant (out of 767!) in Copenhagen on TripAdvisor here:
http://www.tripadviso...penhagen_Zealand.html
This also looks good (and fairly cheap!)
http://www.caphorn.dk/eng/index.html
Reviews:
http://www.tripadviso...penhagen_Zealand.html
If you're going over the Oresund Bridge to Malmo (which is just a short drive away, or an easy hop on one of the frequent trains), I can thoroughly recommend the Indian place on the main square (where all of the tables are outside but, because of excellent heating arrangements, gets loads of customers all year round!). The food is remarkably cheap but still of excellent quality.
Chris