Please note though that you'd probably need to book it as LAX-(JL)-xNRT-(JL)-oICN-(CX)-oTPE-(JL*)-xNRT-(JL)-LAX.
AA used to code-share LAX-ICN with OZ before OZ joined *A, but in any case code-share flights are not bookable for award travel (OZ used to be an AM partner, though). The AA TPE-LAX quoted is also a code-share flight operated by BR. Since you can only have two airlines on a regular AsiaMiles and one of them has to be CX / KA, you would not be able to book AA, JL, and CX on one ticket, unless you use the oneworld award, which would cost you 85000 miles instead.
*It is technically possible to fly CX on the TPE-NRT segment, but that would force you to have an overnight transit in NRT. You cannot have a 3rd stopover in NRT unless you utilise the oneworld award.