I'm super excited for you. My dad has spent a lot of time on the island, I'll ask him if he has any other recommendations.
It turns out that my parents are kind of lame.... :) They couldn't think of much more than some of the standard tourist trap places... I will seem them in person next week and press for some more ideas. In the meantime, they're (kind of tame) suggestions were: Chiang Kai-shek Memorial Park - I remember visiting here.. it is quite lovely. Lots of people flying kites and enjoying the public space. The Grand Hotel is beautiful, and I think worth seeing (especially if you've watched Eat, Drink, Man, Woman). I don't have much more to add... I loved it there. It's a really, really wonderful city.
Hey steve, thanks for asking them! I can only hope so :)