I would also recommend Log 3 gym.
It's near the Panorama near the Hotel de Ville.
The owner, Loïc, is a friend of mine and I know he speaks a bit of English too.
They also have a naturopathy bit to it where they do massages etc.
Otherwise normal gym - weights room, cardio vascular bit, steam room, sauna etc.
You would probably be looking at around 700.- p.a. If you pay in one go and are with the right company for health insurance, you'll get 300.- back (i.e. almost half!)
Log 3 Fitness Center 1800 Vevey. rue du Panorama 12, Vevey, VD, 1800. Téléphone 021 922 82 60