Oh my, sorry for not catching that message earlier!
I am staying over at the International House of Taipei. Even though some websites suggest this place for short stays, they offer half year and full year student accommodations for a very good price. The house includes a shuttle bus to some of the nearby language schools, lunch / dinner box service, laundry options, a small gym, and much more.
As much as the website includes a lot of information, if you like myself can’t speak mandarin (yet), I recommend you write them a mail with your questions and reservation plans. All mails will be responded back in English, so you don’t have to worry about that.
And if you have any additional questions, feel free to pm me!