There are 54 direct and indirect flights connecting Bangkok and Taipei every week, with the journey taking 11 hrs 25 mins on average.
Or, if you're fairly flexible on dates, try to get a last minute-deal by regularly checking the Netflights website.
In Taipei, Taipei Taiwan Taoyuan International (Chiang Kai Shek International) Airport (TPE) is usually the cheapest airport to fly to, but there are one airports to choose from. Not sure who to fly with? Malaysia Airlines, Cathay Pacific & Ethiopian Airlines are the most popular airlines available from Bangkok.
But if you can't find one which is right for you, try looking at flights to other cities in Taiwan to check out some other options.