Copenhagen is always busy on a Friday night and especially the coming three when every single company in the city hosts a Christmas party, so you may not be able to get into the best places, but here are some ideas:
I second the suggestion that you have Sunday lunch at Aamann's Deli - there are other good places for smørrebrød in the city, but they are not opened on a Sunday.
Also as Vanderb rights in his/her nice trip report - Torvehallerne is a must visit.
Finally for a great - and often overlooked Danish delicacy - Meyer's Deli does a lovely Flæskestegssandwich (it's a slice of roast pork with crackling and some red cabbage in a bun).
Meyer's Deli is found among other places in the basement of the department Magasin, where you also find an upscale grocery store and assorted chocolate and bakery stores.
Sorry very late to the party I know, but yes it exists:
Kaffebaren - Æblestien (Apple Lane) 2 - no website but a facebook-page: http://www.facebook.com/pages/Kaffebaren-pa-Amager/125281017511369
In the newly developed Ørestad part of Amager you find Foodshop 55 - great bread, coffee and pizza:
Aamanns is a must try - but in my opinion Ida Davidsen is far from the best of the old traditional smørrebrøds places - it's not bad, but it's just not the best.
Ambiance is a huge part of a visit to a frokost (lunch) place, and Davidsen is just a little tired and dinky.
My favorite is: