Easiest and probably cheapest is to just take NW via DTW/MSP and NRT.

Sometimes it's a bit cheaper to route with an extra stop via the west coast between NW's midwestern hubs and NRT (usually the stop is in LAX or PDX). If you really want to juice more miles out of it, you can also route via HNL--which sometimes knocks the price down even more. On the way to SIN, routing via HNL requires an overnight. But on the way back, you can get a half-day in Hawaii without the need for an overnight.