Although I don’t live in Taipei, I know of one school that can probably help you. It’s called the Taipei Language Institute. It offers courses pretty much any time that you want. They have branches in Taiwan’s major cities. Last I heard, as long as you can round up enough students to start the course, they will start one for you. To round up other students to start study at the same time as you, it would be easiest to post on an expat community forum such as Forumosa (Taipei) or Kaohsiung Living(Kaohsiung). Also, I think that if you were to contact them in advance, they could help to place you in a class that hasn’t filled up yet. I’m sure that beginner classes, being most in demand, start up somewhat frequently.
One thing you may want to consider when choosing your school, is what you want to learn. Do you want to read or read and write? If you want to learn to write, it will more than double your workload, as characters are very complex and only learned by rote memorization. Most Universities only offer writing/speaking courses.
It’s my understanding that TLI, however, focuses first on speaking (to help you learn to get around Taiwan) and then moves into the characters. I have never studied at TLI, but know many who have. I’ve never heard anyone say it was spectacular (but learning Chinese is never spectacular). I have never heard anyone complain about it though either.
You have one more option, with regards to school. Thats to apply for a sixty day multiple entry visitor visa. This visa is good for 60 days, and can be extended three times (six months). Then, you could stay in the country without a student visa, and study with a private tutor. Most I know charge about NT$4-500 ($12-15 USD) per hour.
I would also recommend, of course, that you both start studying before you come. I’ve found the Pimsleur language tapes to be very effective.
I’m sure that if you do a bit of searching you can find some helpful free websites online as well.
Good luck with your studies.