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.
You can use the filters on your search to see more flights and find cheaper rates – Taipei has more than one airport. So if you don't need to fly into Taipei Taiwan Taoyuan International (Chiang Kai Shek International) Airport, it's worth checking your other 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.