If you want a cheap flight from Los Angeles 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.
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.