Every district in Taipei, has a membership free gym. You pay by the hour, for 50NT/hour. Of course, these are public gyms and public gyms, you get the public gym etiquette.
I would probably stay away from World Gym. Based on friend's reviews and that on-going thread in the forums, not a great experience.
You can also try out Formosa Fitness over near Guting. He also has a branch down in Xindian and was thinking about another, not sure where. Foreigner owned and specializes in kettle bells if you're into that. He does do a monthly membership. Not sure if he still gives a Forumosa discount, but doesn't hurt to mention it.