Cafe Trouve, another old colonial-era Japanese house turned espresso cafe. Not that it's a bad thing. They each have their own appeal and comfort. This one is near my school at Shida (師大). It's not in the night market but close by it and definitely walking distance from it. As Yuki mentioned, it's an old wooden Japanese house used as Taida (台大大學) faculty housing before and in May 2011 was converted to an espresso cafe. Although you wouldn't know it unless you looked up or at some of the support beams. The old logs of wood criss-crosses everywhere to form the skeletal structure of the roof. What's unique about this place, is the small antiques store at the back of the store. Walking through it, they're mostly antique oddities from the United States, Europe or Russia (yes, russia). I found that odd to find russian antiques in Taipei. They also have a few black & white photos framed on the wall to enjoy. The coffee was great and so are the fruit slushes. It was a hot day, so slushes and A/C was just comforting. Coming to this nice cafe though, expect a wait time. We lucked out at 2 minutes, but it was pretty packed. On the other hand, it was a sunday afternoon and 4 tables were reserved for a large party. Prices run around the typical Taipei cafe prices of $150-200. Here's the location and website info below as well as a google image from Catherine Shu of the Taipei Times.
Cafe Trouve (找到咖啡)
No. 4, Lane 16, Tàishùn St, Daan District
Taipei City, Taiwan 106
Sunday - Thursday 10AM-11PM
Friday - Saturday 10AM-Midnight