If you enter Wenzhou Street from Xinhai (or walk north), just south of a small shop decorated with hanging CDs you'll find Cafe La Boheme on one side, and Coffee, Tea or Me on the other. The placement somewhat mimics the placement of Cafe Bastille and Shake House down the road (running along Lane 86, catty-corner to the Lutheran church where you often see a guy who walks his Persian cat on a leash). Those cafes are both great - Shake House is better - and so are these two.
Shake House is one of my all-time favorites, with friendly staff, an artfully shabby space, usually decent music, great beer and, for a student beer cafe, pretty good coffee and food. Bastille has a snotty staff and horrid food - though the focaccia sandwich is OK - but good beer, plugs and Wifi. You can use Bastille wifi from Shake House, but there are no plugs.
Anyway, back to the two cafes at hand.
La Boheme, on the west side, has a velvet-furred declawed tabby named Luna, a selection of books in English and Chinese, generally good but more popular-music oriented music, beer (Belgian) and excellent food. I mean truly good: their burgers are stupendous, generous and well-made. The burger choices are unexpected: I recommend the burger topped with apples and white wine sauce.
Their fries are just as fries should be - not too salty, crunchy on the outside and soft on the inside. I've also had their herbed lemon chicken, which was quite good. Their hot chocolate is absolutely excellent - dark and chocolatey, you can smell it before it reaches your table. It's rich and tasty, nothing like the wimpy brews of the various "chocolate" cafes found in Taipei. If you love a good hot chocolate, go here and nowhere else. It's more like drinking chocolate than "hot chocolate" (which in my mind conjures up Swiss Miss).
It has its own wireless network and plugs, and is very "local" (foreigners do come, but they're not the norm). With green and yellow walls and a wood-floored raised area, this place is more brightly lit and comfortable like an old chair.
With its decidedly less healthy (but no less tasty) fare, it's Gongguan's answer to Zabu, whose tasty Japanese food feels downright good for you, despite the many fried menu items. That's Japanese food for ya.
It's rare to find cafes in Taipei that have a good atmosphere and good food, let alone genuinely good hamburgers, so this is something to take notice of.
Coffee, Tea or Me (also called "Cafe, Tea or Me") has two, possibly three cats. It's hard to say and only the orange cat is friendly. You have to steal La Boheme's wireless, but they have their own plugs and truly excellent coffee. La Boheme's coffee isn't bad either. They have more space, very odd books and other decorative bits and bobs, more chairs but a limited selection. You can get drinks of all sorts, from good coffee to a very limited beer selection - really just Erdinger, a few boring choices like Heineken and a vanilla-y French beer in a blue bottle. They also do cocktails, and have an impressive bar for what is basically a coffeeshop. They made me a drink at their suggestion: Jameson, some sort of bourbon and Grand Marnier with ice and I assume something that was not alcohol (or maybe not - coulda been pure alcohol). The serving was generous and it cost NT$140. WAY TO GO.
Coffee, Tea or Me's space is more artfully distressed, with dingier walls, a menu board that's impossible to read, hodgepodge chairs and tables - including one low set of chairs in blue faux velvet and another pair of antique-looking Chinese style bamboo wicker chairs. It has a great atmosphere but is not the place to go if you think you'll get hungry.
I strongly recommend both, but for entirely different reasons!