Doesn't seem much point if you'll only be here for a month! Take a break
Frankly, you're not going to find a gym anywhere (I've tried in various different countries) that will sign you up for less than a year.
If you change your mind about leaving, I recommend going to California Gym at Taipei Main Station. It's only 15 mins on the MRT and it's all top-notch kit. Been going there for years.
Just remember: don't accept the first price they give you, which will be outrageous. Ask for a cheaper, limited membership plan. If you can stay a while (a year or more), tell them. I'm restricted to ONLY that particular gym and I pay 1300 a month.