Choosing between Kaohsiung and Tainan is tough, I personally would spend 1 night in each city and feast on all the street food in Tainan.

Taichung is a little too cookie cutter for me. But I see you're using it as a stop after Sun Moon Lake which is a wise choice.

The drive between Alishan and Sun Moon Lake has some gorgeous views of Jade Mountain (highest peak in Taiwan), but do be cautious in bad weather. I took a bus trip on that route during the pandemic, and we were the only passengers on the entire journey. I missed those quiet days.

Finally, I see that you missed the Presidential Elections on Jan 13 by 3 days. As a foreigner living in Taiwan I must say the campaigning right up to election day are sights to behold. Hopefully you guys get a good glimpse of the democracy we cherish so very much while you're here.