Day 1, Arrival (evening liao) -> Taipei -> Check-in & travel to Shilin for dinner + shopping (if leave shilin by 8pm due to boredom, then travel to ximenting (muchuan ming ge chanting) to listen to music (esp. during wkends)
Day 2, KeeLong for late morning meal -> Yeliu (late morn to afternoon) scenic tour (seafood lunch) -> Danshui in late afternoon for hawker food + sunset. After tht, can go to danmu for shopping
Day 3, Travel early in morning to JiuFen. Come back after lunch and prepare to move out of Taipei. Have to decide whether to go east or west taiwan from here. Lets assume go west taiwan. Then travel to TaiChung 1st. Check in hotel and visit the local night market there
Day 4, Leave v. early to go to JiaYi to catch small red train to Alishan. Spend the rest of day exploring alishan and rest early for night
Day 5, Wake 4am to catch sunrise. Prepare to leave alishan after that ~11am. Travel to Kaoshiung to explore the city and the night market.
Day 6, Continue to explore the city sights. After lunch, travel to TaiTung. Check-in evening and explore city at night
Day 7, Explore sights around Taitung. Have to travel to Hualien in afternoon. You'll arrive v. late in Hualien. There's some sports like river rafting which you might want to +1 day to enjoy.
Day 8, explore hualian + Taroko Gouge.
Day 9, After breakfast, travel back to Taipei. If you're self-drive, can explore the coastal sights on the way back, which will take u 1 day to do so. If not, you'll arrive taipei pretty late afternoon. You might want to visit some of the touristy area like CK memorial hall, sun yat-sun temple etc + red light district etc.
Day 10, Prepare to leave Taiwan