If none of your stops are > 24 hours, I am guessing one SWU should last you BWI-NRT (UA doesn't fly NRT-KIX) then another for KIX-BWI.

Since you've already gone thru NRT once on your way to SIN, I am not sure whether UA will let you use the same SWU for the HKG-NRT segment. I know that BWI-HNL-SIN-HKG on 1 SWU would be fine because I've done something similar before.