For a very thorough discussion of the topic of visas, take a look at Oriented's Legal Matters Page. You might also want to read this article.
Several universities in Taiwan have a Chinese Language center where foreign students go to learn Chinese. These programs will allow you to extend your visa indefinitely, as long as you stay in the school and maintain a good grade average. After a few months of study, they will help you apply for a work visa. There is also a generous scholarship program for
diligent students.
Chinese Language Centers
National Taiwan Normal University, Taipei
National Taiwan University, Taipei
Wenzao Ursuline College of Languages
Also, try this page for a more extensive list of Chinese language centers in Taiwan.
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