There is train from the airport to the center of Copenhagen and from the center of Copenhagen to Klampenborg Station (100 meters from KKKK - our kayak club). Plan your journey in the Copenhagen area: http://www.rejseplanen.dk/
Links to nearby accomodation:
Camping 4,2 km from KKKK: http://campingcopenhagen.dk
Hostel 4,4 km. from KKKK: http://lyngbyhostel.dk
Hotel 2,4 km. from KKKK: http://www.skovshovedhotel.com/
Good offers on hotels in Copenhagen C: http://www.hotels.com
Homestays may also be possible.