I would look at using CI (China Airlines). You would probably connect through LAX and TPE (Taipei). You will find pretty good availability that way. If not, look at getting to NRT, then changing to CI. Or Shanghai via China Eastern, then on to TPE. You can also redeem on MH (Malaysia Airlines) via NRT and KUL/BKI.

Also, remember that KE has huge black out dates, so you will not be flying through Seoul any time this summer.