Japan Airlines has Los Angeles to Taipei TPE at $482 round trip with one stop itinerary via Osaka KIX. There are some departure dates in May at $505 round trip and then August to January 2018 has $482 ticket prices, except December.
Japan Airlines Los Angeles LAX – Osaka KIX – Taipei TPE – Osaka KIX – LAX
I show Google Flights here for display purposes only since Expedia and JAL do not show flight numbers on a simple one page image.
JAL fare rules for this economy special fare:
ECONOMY CLASS — LIMITED TIME OFFER / RESTRICTED
No restriction/3 days/5 days/7 days/10 days/14 days/First Saturday (*1)
1 month/2 months/3months/4 months/6 months/1 year (*1)
ADVANCE SEAT SELECTION
S class: Permitted; N class:Not Permitted
RESERVATION CHANGE FEE (*2)
CANCELLATION FEE (*2)
JAL MILEAGE ACCUMULATION
S class:50%, N class:30%
JMB UPGRADE AWARD
2 permitted at US$100 each