You can run or jog just about everywhere in Copenhagen. Check the park articles for more information, and the recreational page for swimming, skating and biking. In the parks you can play ball games, with Freesbee or your pet animals.
Kayaks are available in the harbour near the Parlament and the old stock exchange. http://kayakbar.dk/
In the end of August there is a swimming race for 2 km around the parlament. Christiansborg. You can sign up here
There are a lot of Fitness centers all around the city. The biggest is Fitness.dk . Note that they say that you can get a membership for 100 kr. But if you want the full package, it is 250 kr. a month.
FC København is the biggest soccer Club in Denmark, and they play at the national stadium. So if you like soccer, this is the offer. https://www.fck.dk
Brøndby, south of Copenhagen, is the biggest competitor. When these two teams clash, it is really a party https://brondby.com
In the month of may – Copenhagen is co-hosting the world Championships in Icehockey. See more here http://2018.iihfworlds.com
Copenhagen marathon is in the second weekend of May. It is a Sunday and the citycenter will be closed for car traffic. It is big a party, with a lot of people cheering and bands playing. The same goes with the Half Marathon at September 16th.
If you are completly out of your mind, you can also join the Ironman Triathlon on August 19th.
The Local Roller Derby team is CRD Check their Facebook here for dates.