Thanks all for your words of advice. As an update to my saga...

The PHT was booked solid. Absolutely no availablity within a week before or after I wanted to be there. Because of a mix of timing, money, and people's advices from here, I am going to go to Taipei for two nights, but because of the way I am planning the flights, I will have almost three days there. Given a choice, I would rather go somewhere that I might not go to under normal circumstances, but I figure that flights to Tokyo are cheapish, usually, and that I can save it for when I have the $$$ to be able to really go and enjoy the city.
Thanks again for all of your help.