A great way to relax in Taipei is to soak in healing mineral waters.
The fact that Taiwan has some of Asia's finest hot-springs is well known; less known is the fact that you don't even need to leave Taipei city to soak in some of the island's best springs. Taipei's Beitou (the historical spelling is Peitou) is where locals come after a hard day at the office to soak their cares away in healing, sulphurous waters and still be home in time for a late supper.
There are plenty of hotels and spas offering private pools and even massages, although some of the ritzier ones can be costly. If you want a more down-home experience, check out the outdoor public bath inside of Beitou Park, Millennium Hot Springs, where you can enjoy a soak under the sky with the general public. Bring a bathing suit.