Just wonder if you have considered a connection in Tokyo or Seoul? Those do not add much time to your itinerary over a SEA stop-over. JAL, ANA, etc all has very good seats and service for their flights to JFK.
There are 3-4 flights an hour (~50/day) TPE-HKG on CX, KA, BR and CI. But no discount carrier on this route. You can just go to any travel agent in Taipei to get your tickets. Prices should be competitive between agents.
Only possible alternative, if you really need to save money, is to check on flights TPE-MFM (Macau). They may be cheaper, may be not. Add the cost of the ferry (~HK$150 one-way) to Hong Kong. Will add to your travel time significantly.