--yes, sufficient time from Taiyuan to Chiayi with the high speed rail
--since you've quite an active schedule anyway, I'd make a point of getting to Tainan
--Tainan over Taichung, definitely
--personally, I don't get Sun Moon lake
--If you're going down to Yilan for whale watching, possibly you could then go on to Hualian for the night, and then both beaches and Taroko Gorge are 1/2-day excursions.

just some ideas.