Here are the only 2 I know of, so far:
Taipei Tourism Office - in the underground mall between chung hsiao/fu hsing and chung hsiao/tun hua metro stations. Stays on all the time, so if you go there and it’s closed you can sit at the plastic tables down the hall for the gov’t employment agency and connect.
IS Coffee - next door to Hooters, just off the nan jing/fu hsing intersection. I’m not sure if this hot spot actually belongs to IS or the neighbor. It was labled ‘EzLink’; the girl who was working there said that branch did not have wireless, but it did and I was able to connect with my mac. A bit chunky, but it’s free.