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.
You can expect accommodation prices to be higher at this time of year too. Last-minute deals do crop up, but it's best to book everything in advance to avoid disappointment.