The following just dawned on me:

1. OP's situation is extremely well suited for a 100k mile BA redemption (biz class RT) because LAX is a gateway for both CX and JL and tends to have better availability than other gateways (though, maybe not now for JL since the 747 is gone)
2. OP should get apply for the Chase BA card ASAP (deadline could be tomorrow, though many people believe the offer will be extended); check out the monstrous thread in MilesBuzz! for more info
3. Even if OP's miles don't post in time for trip, I'm guessing a forward trade (with someone who got in on the offer sooner) could be arranged
4. Try to get J seats on CX (to HKG) or JL (CAN or HKG)

Step 4 will probably require a certain amount of flexibility/determination, but the ensuing experience should justify the time/effort expended (doubly so, in light of the fact that the OP's out of pocket expenses would be limited to a small amount taxes and he/she would get to avoid CZ).