There are zero 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.
Taipei Taiwan Taoyuan International (Chiang Kai Shek International) Airport is just one of the airports in Taipei. If you're not set on a specific airport, try changing your search from Taipei Taiwan Taoyuan International (Chiang Kai Shek International) Airport to Taipei to see a broader range of flights – they might be cheaper.
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.