NRT-SIN-HKG-xSIN-LHR-FRA-JFK-SFO-SIN-NRT
Let's see.... I would change the last few segments, and go down to a 29,000 miles fare. Rather than JFK-SFO-SIN-NRT, I would fly JFK-LAX-NRT, or JFK-SFO-ICN-NRT. You would then have a few thousand extra miles to play with.