Lucky you! We went to Copenhagen for three years, we thought, and stayed sixteen! Would recommend Hellerup, just outside the city and by the coast (beaches, daily swimming, fresh air!) There are two international schools there, the Copenhagen International School (CIS) u p to 18 and Rygaard, a Catholic international school which goes up to IGCSE only. Both good, our daughters went to CIS, one from 4 upwards and loved it. They did the international baccalaureate curriculum there, which is excellent, even for little ones. Ask them about the waiting list, I don't know the current situation, they are both v popular.
Denmark is a favourite expat and UN place to live and there are some other international schools although no British ones only. You might prefer to be in the city itself or in Frederiksberg or Gentofte and of course it depends on how far you or your husband have to commute. But commuting is pretty easy, with trains, tubes and most importantly by bike in safe bike lanes with their own pavement! A great place for children and a peaceful interesting life style. Have a lovely time.