Great points, everyone. Okay, how about:

DEN-LAX-SYD-BNE-ADL-PER-SYD-HKG-NRT-SIN-BKK-HKG-DXB-LHR-DME-MAD-LHR-YVR-JFK-MIA-DEN

This improves mileage to 46,515.

We'd be in CX First for the SYD-HKG-NRT and BKK-HKG-DXB.

I agree that it makes more sense to use the Shinkansen to return to NRT instead of wasting that leg.

It's true that LHR-DME-MAD-LHR is not ideal, but haven't come up with anything better yet. There is a limit on flights between LHR and various destinations (Middle East, Russia) that causes routing issues. I'm open to suggestions.

websterlewis, I agree that a lot of the trick is getting close to where you really want to go, as comfortably as possible. We will definitely be taking some side trips by train and short-hop air throughout this trip.

Thanks for all your help so far!