There are tons of good coffee shops in Shida, but Mo!Relax is one of my favourite. Because it's not just a place to enjoy a good cup of real expresso, but a kind of public living room. They've got an extensive library (books, magazines and comics), a good selection of CDs, videogames, and, of course, free WI-FI.
Don't forget to try their delicious gyoza.
MRT Taipower Building (Red-green line)