A family home: convenient, comfy, and designed for the whole family
The good cafés and shops at Sønder Boulevard/Dybbølsgade as in 'Kødbyen' and nearby 'Enghave plads'/Istedgade
AbsalonCPH, a social place with a lot of interesting things going on, where you can eat very good and cheap food at longtables (http://absaloncph.dk/)
The old buildings and stone elephants at Carlsberg with the little park behind the café 'The Elephant'
The 'Enghave park' (with a rose garden and a good playground), the park: 'Søndermarken', the Zoo, the park: 'Frederiksberg have', 'Skydebanehaven'...etc.
The Swim Centre DGI (http://www.dgi-byen.com/the_swim_centre/)
All the other good cafés, restaurants and shops, streets, museums, parks, nature sites, the sea, Christiania, Tivoli, The Blue Planet, concert places, theatres and not least good and green bicycle lanes (and city bikes)... in Copenhagen
And the good things around Copenhagen... as the Louisiana art museum, Arken, Bakken (http://www.bakken.dk/english/bakken-the-worlds-oldest-amusement-park), Dyrehaven... etc. etc. :).
To organize a home exchange, you need to be a member