I'm trying to think of a sleeve that you could fit onto the the existing damaged pull, on the order of, but much larger and thicker than one of those spines that slides over a folded 17" x 11" sheet of clear plastic that serves as a protector for sheets of paper, like for a written report or whatever for a school assignment. 

I feel like I've seen something like this in home improvement stores, but can't think of the department or its purpose.  Probably a plastic trim piece for cabinetry or whatever.  If such an item exists, you could cut it to size and slide it on, or use some JB Weld to attach it permanently.  I think there's enough left of the broken off section to allow a scheme like this to work.