Are there ANY gyms at all in the Mucha area? Wen-Shan 文山 area?
Unfortunately no. But with all those mountains near by, plus the university I don’t really miss one. I jog on the dikes, cross over to the university, jog through there, do chin ups and dips at the track (they have the bars there), and then head home. A good overall workout.
If you are willing to wait another couple years Mucha will have a rec centre, including gym and swimming pool like the ones on Jungshan North Road and Beitou. The Taipei city gov is building one gym complex for each of the 13 districts. Hope your gut can hold out (or in) that long. 
I thought there was a uni gym you could join?