Why oh why can't it be white?!!?

I love this shot, I'm happy I got it. It's a good picture.

But I didn't bother submitting it. What I wanted, what I know *they* want, is the background to be perfectly, completely, utterly white. I can get it looking pretty good with some time and effort in Lightroom and/or Photoshop, but I want to do it in-camera. And I want to do it with my 2 SB-600 Speedlights.

What I'm trying so far is putting the camera in commander mode. Camera is in manual, I'm choosing 1/125 - ish shutter speed and varying f/stop to taste.

1. On camera flash set to -- or TTL -1.3 (trying some use as a fill flash).
   
2. One on the floor aiming up at the background, behind my subject, set to TTL +1.
3. One behind an umbrella, camera left or right, aimed at subject, pretty close, set to TTL 0.

Maybe I need to take 2 and 3 up a notch? Maybe I need to go to manual mode on the flashes?

I resolve to spend some time on this tomorrow and at least get it better.

(Putting the flash directly behind the subject aimed straight back seemed like a good plan, but the CLS was iffy with no direct line of sight to the camera...)