So finally, I found the perfect spot: Think Corner (Tiedekulma), which is located in the city center -just behind the main university library-. This space belongs to the university also somehow: It is kind of a cultivator for entrepreneur minded students' ideas. A place for inspiration.
The building includes study space, some beanie chairs to chill, a small cafe with delicious food and reasonably priced coffee, some office area, a common kitchen which is open during weekdays and events, and and... there is a big UniSport facility, the Kluuvi branch.
I did enough advertising I believe. Come to one of Think Corner events or just for a quick peak and check the place yourself. I guarantee you won't regret it. Maybe see you tomorrow there!