In 12 days you could do a full lap of the island if you wanted to, either by self-drive or a combination of HSR/train/bus:
Taipei, Sun moon lake, Kaohsiung, Kenting, Taitung, Hualian, Yilan/Jiaoxi, back to Taipei
Not sure what your dates are but note that April 3rd and 6th are national holidays in Taiwan, so anywhere outside of Taipei will be extremely busy, traffic heavy, hotels expensive, and train tickets hard to book.