There are two airlines offering flights to Taipei. The most direct route isn’t necessarily the cheapest – being flexible is key to finding the lowest airfare rates. For example, try searching by month to see the best time of the week to fly. The cheapest date is currently 02 / Apr / 2019.
Make sure you use the search options above to search for flights to other airports in Taipei, not just Taipei Taiwan Taoyuan International (Chiang Kai Shek International) Airport. The more options you look at, the better your chance of finding a cheap deal. Just so you know, the flight typically takes 20 hrs 25 mins.
There are about 42 flights via Taipei every week, so you have plenty of options.
During this peak season, accommodation prices often increase too, so it's best to book early. Or you could keep your fingers crossed and hold out for a last-minute deal.