There are a few places to check out but they vary in quality.
Best to check out this link n a Friday, as they publish what's happening about town in music for the next week.
check out the entertainment or events section:
http://www.taipeitimes.com/News/
There's the blue note, there's the riverside cafe etc...but check this one out - once in a while they have good players, and they also have jams:
http://www.sappho102.biz/
Nearby there is a cafe called 'friends' or some such which also has jams of a jazzy flavour.
There are a lot of really good players in Taipei, particularly world class studio drummers. Also many session musicians from around the region record there because its where most of the mandarin recordings are made; and some times they come out to play....
Taiwan is no fun to shop anymore for MI, it used to be 10 years ago when you could still buy OEM items, at very good prices... but you can still get quite a few things new or used if you check out all the shops. FYI most of the manufacturing is not in Taipei but further down south.
