I normally wouldn't recommend Delta, but I have found the availability to TPE from LAX on China Southern (CZ) wide open, especially in business class in their new A380 using Skymiles. The route is LAX-CAN-TPE......the same route back. I found the availability for CZ from ExperFlyer.
Do you normally travel alone? If planning in advance and you only need one seat, coach or business, Delta usually have just one seat per day per cabin going from LAX-NRT-TPE.