I suggest you try some of the cosy sandwich cafées you'll find in Copenhagen. Many of them actually serves a sandwich you'll get satisfied on for a couple of dollars. This is a picture from one of the small restaurants or cafées you´ll find on many places in copenhagen. I remember I ordered a big sandwich with plaice, shrimps, salat and other delights among a tuborg and a cup of coffe for 25 Dkr(around 2,50€ or 2£) when I visited the small shop Klareboderne in 2014. I beleive you are still able to get a delicious roll(rundstykke) with cheese and a cup of coffee for the same price.
Some of the wonderful sandwiches they offer on different cafées offer here. You'll find a hundred of these cafées in Copenhagen. The easiest way to find them is by searching "smørrebrød, København, Danmark" in google maps.
The most famous of these cafées or sandwich restaurants is propably Ida Davidsens café that started as a wine bodega in 1888. This was a complement to an Oscar Davidsen winestore where customers could come in to taste the wine he where selling. Soon he discovered that the customers also where hungry when they came here so his wife, Petra, got out in the kitchen for making some delicious sandwiches. His wife where creative and a good chef which gave the result of 141 different sandwiches who became very popular along the guests. Soon the small café where crowded during lunchtime so they where forced to find other places.
In this café, as many other cafées, you are able to buy a couple of favorites. Like a sandwich called " the vets night sandwich". The name came from a famous vet that where hungry during the night and, therefore, went to the café to get a sandwich made from danish rye bread, liverwurst, salt meat, fat and sky( a kind of gelatined buillong). I think the name sounds funny because it sounds like the vet is making food of his failed patients during the night but it is actually a vet named Sigurd Keilgaard that invented the sandwich. Many cafées also add some raw red onion or fried yellow onion on top. You could also add some pickled cucumber and persill on top.
Danish open sandwiches by Mette Herborg