I would try very hard to avoid PEK on this routing, lots of delays there and particularly since Air China is not exactly a great experience. There's ANA routing via NRT June 5-23 for $1088. ANA is better than Air China and NRT better than PEK and more reliable for schedules. Maybe look into Chinatown consolidators and see if they have anything reasonable for nonstops LAX-TPE on EVA or China Airlines. IMO, it would be worth paying a reasonable per ticket premium to go direct and avoid a connection at all. If they are flying from ORD (per your profile location), then try to fly ORD-Asia without connecting in the USA (especially LAX) unless you get the deal of the century.

Since that's high season, you don't want to wait too long, but I'd keep digging until first week of April then pull the trigger, see if any "spring sales" appear which might at least drop the outbound fare class a bit.

Oh, and Welcome to FlyerTalk. Don't be shy about posting.