If you want a cheap flight from Manila to Taipei, you've got zero airlines to choose from.
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.