Photoshop Elements :: How To Achieve True White Background

In the attached image, I'm trying to achieve a true white background in PSE. I put a true white box in the image to show the gray background that's in the original shot. (I understand it'd be best if I could achieve the true white with the camera in the original shot, but this seems to be the best I can do with the camera. I set white balance and over-exposed as far as I could without washing out the subject. Still a gray background.)
 
The problem I'm having is that every PSE tool I use to try to adjust the background image washes out the thin hairs of the grain. Ideally, I'd like to be able to create a perfect mask so I can just kockout the background. That way, I can adjust the tones of the subject without affecting the surrounding area.