Your best bet is to look at places on AirBnb. There are a few hostels that have listings on there that are pretty cheap. I recently found a website for a place called InnApartment that looks pretty good too (innapartment.weebly.com) and I'm planning on staying there when I visit in March.
But overall, Taipei's accommodation websites are local-focused and most sites are going to be in Chinese, which makes it hard to find places online.
Similarly, the coworking spaces are more focused on local startups and creatives, and again, most sites are in Chinese. Though I've heard of an up-and-coming coworking space called http://futurewardcentral.com/ and the founder seems to want to attract more digital nomads. There are also a few other options but I haven't checked them out yet.
On the plus side, Taiwan has amazing infrastructure, fast internet, and you'll be able to work from anywhere as it's pretty much the coffee shop capital of Asia.
Oh and the food is other-wordly too. So more pros than cons