We stayed three nights with our children (4 and 7) at Natalie's appartment which was just the perfect place to discover Copenhagen !
Natalie was very welcoming, showing us everything and she had also left some delicious bread, juice and food in the kitchen for us. We really appreciated those delicate attentions. The children were allowed to play with the toys which made them very happy.
The apartment is SO beautiful with design furniture and very spacious; the neighbourhood is very pleasant too with all the shops, restaurants and playgrounds for the kids.
You can have a nice walk to the city center by the harbour (about 20mn).
I would highly recommand you to stay at Natalie's !