I have been two times to Copenhagen, it is only 300kms so approx. 3h/4h by train from Linköping (they don’t have fast train yet...)
We stayed on Airbnb (cheap, well located, with bikes)
Day 1 : We arrived around lunch time and we went to Nyhavn to have lunch and try the famous Smørrebrød (sandwich basically), it was really warm and we decided to walk in the center close to the sea. We had an ice cream to Vaffelbageriet
Day 2: Brunch at café G, and we bike all day to visit the little mermaid, different churches, parks and I even swam in the harbor! Too warm! We looked like 2 tomatoes at the end of the day.
Day 3: We visited parks, had a brunch and it was time to go home!