Personally, I would try to avoid CI. Take DL LAX-NRT-TPE or use the LAX-HND flight if availability of award inventory is a problem. You could also try SFO, PDX, or SEA flights to NRT on DL. You could connect through NRT again to HKG, but this flight arrives late at night so I would try to find an alternative SkyTeam connection, perhaps CZ through PVG or CAN with visa free transit. For the return, I love the HKG-DTW 777 nonstop, although it surely isn't a legal routing to LAX, so you could do HKG-NRT-LAX or the analogous alternatives as suggested for the outbound.

If your travel dates are not blacked out by KE, look for LAX-ICN and then connecting to TPE and HKG. KE is a nice airline and ICN is a wonderful airport, better than SIN or HKG except for the lounges.