Coworking is still a new concept in Taipei, but there are a few spaces. Most of them were formed as business incubators rather than pure coworking spaces.
BECC Consulting is a consultancy which also lends out their extra office space. The location is in Taipei's laid back Da-an district with a nice patio, whiteboards, and a pretty good sized meeting room with projector whiteboard.
Location
YSeed is a business incubator/coworking space in Taipei 101. YSeed is the most well organized and formal incubator program and has organized startup events in the past.
Location (Taipei 101 37th floor)
IEHub@NCCU is another Incubator and startup oriented coworking space located at NCCU university. Rent is around $2000/month per person.
Appworks Taiwan (chinese only) is an incubator/coworking space with 5 meeting rooms, an espresso machine, 24hr access, and regular office amenities like copier/fax machine/land line phones. However, just like YSeed, you have to apply to use the space.
Location
Most of the incubator-type spaces require applications to get in. but they will help you secure funding if you give them a good pitch. IEHub and YSeed speak English. Appworks is chinese only.