Everyone is different, but I have a feeling that most travelers would not want to spend 10 days in Taipei. If you have the money, traveling by HSR (High Speed Rail) makes rather long journeys quite practical. Be warned that you should NOT attempt to charge any tickets on the HSR as they don't do real credit card charges. It will come as a cash advance against your credit card and you'll be charged interest on the purchase daily until you pay off the entire card balance.
Tainan is quite far south, but the HSR makes it a practical day trip. It has a lot of old temples and buildings. Note that the HSR station is outside of the main downtown area, so you may want to take a taxi or figure out how to go by bus to get to the main downtown area.
Kaohsiung is the last stop south on the HSR and I don't recommend going there. It's not a bad city to live in, but it has little of interest for tourists.
Sun Moon Lake might be of interest to you, but I must warn you that if my Taiwanese friend had not gone with me, I probably would never have been able to get there by myself without being able to speak Mandarin or Taiwanese. We went there from Taichung by bus. Here's some info on getting there from Taipei, which might be better than what we did.
http://wikitravel.org/en/Sun_Moon_Lake