score:8
There are many, many internet cafes, but most places hardly advertise. They typically run 10-35NTD an hour. Your best bet is asking your hotel/hostel staff for 網咖 (Wangka), or if no one understands you, 電腦咖啡店 (Diannao Kafeidian). You can even just ask people on the street (I've done this many times when I was new to Taiwan... just repeat Wangka until someone points). Taiwanese are typically very helpful.
You can also print out the following and someone will point you there:
對不起
網咖在哪裡
謝謝
Upvote:2
Beitou Public Library, located in Qinshui Park, has public internet terminals. I'm unsure if they're free or not though, but you don't need to be a member.
Randomly, Lodging-world has a query that lets you search for hotels with wifi in Beitou. So if you're going to stay at one of them, then you're fine, but if not you could go along and see if their reception area has free wifi for you to use for a while. I did this in a hotel in Tashkent, Uzbekistan - ordered a drink and sat in their foyer, and they were fine with it.