I am not sure what Krispy Kreme does to recycle. I read that NC State University does a Krispy Kreme Challenge, where they recycle the boxes and any doughnuts that are leftover they will be donated or composted. The university also recycles the shrink wrap that comes in the box or when they give a single doughnut. If you think about it, the boxes can be reused because they have shrink wrap in them so the box does not get dirty/messy from the doughnut.
https://sustainability.ncsu.edu/zeroing-in-on-waste-at-krispy-kreme-challenge/