I've been to Berlin but I haven't ridden a bike there. But looking at the map you should definitely check out the "Tempelhofer Feld" which is the old airlift airport that's open to the public nowadays. You can ride a circular route or do sprints along the runway. good fun.

Apart from that I think if you head out of Köpenick to the southeast that there should be some good riding - it's not called "Müggelspree-Löcknitzer Wald- und Seengebiet" (Forest and Lake District) for no reason :)

Cheers and enjoy your time. And don't forget to explore Berlin, there is a ton to see and 10 days of after-work free time won't nearly be enough!