Hi
Monbijoustrasse is great. It is close to Aare and also close to the city center (ok, in Bern, anywhere is close to the city center..).
Anyway, you could go swimming to the place called Hallenbad Hirschengraben at Maulbeerstrasse 14 (this street is adjascent to yours): http://www.berninfo.com/de/navpage-S...BET-33544.html
And there is a fitness center at Effingerstrasse which is going to be even closer: http://www.fitness-plus.ch/
Hope that helps :-)