We recommend that you book your room at the Zleep Hotel in Ishøj. It is not a posh 5* hotel, but it is good and cheap and it is very easy to get from there to the Kennel Club building. The hotel is in the same building as the S-Train station and train E towards Køge will take you directly to Jersie, where you can see the Kennel Club building from the station. It takes 10 minutes by train to get from the hotel from the Kennel Club.
It is also very easy to get from Copenhagen Airport to Zleep Ishøj. There is a train going directly from the Airport to Copenhagen Main Station (København H). From there you get on train E towards Køge and get off in Ishøj.
Book your room at Zleep Ishøj here: http://www.zleephotels.com/en/hotel/zleep-hotel-ishoj
Time table (train) from København H to Ishøj and from Ishøj to the Kennel Club: http://www.dsb.dk/global/pdf/koereplaner/afgangsplakater/linje%20e.pdf