ok, thats a bit of a hard question, there is so much to see and do

If you are into adventure type stuff, Queenstown is a definite, lots of bungey jumping, white water rafting, jet boating etc to do
Christchurch and Dunedin are my faviourite cities, I havent been to Invercargill, but they do have a indoor timber velodrome and the people from down there are very friendly
There is also plenty of national parks if you want to do some walking, all with great scenry and facilties, ie milford sound, frans joesph, see photos below
A very nice ride would be from Christchurch, down past lake Tekapo and through to Wanaka. Would take a few days but is very senic. Most people tend to go down the east side and then come back up the west coast, the west coast is wild and rugged with not so many people
Still got plenty of time to get fit for Copenhagen, see ya there