Step 1: Busted!
Sugru to the rescue! I can use it to fill in the missing chunk and hold everything together, well enough to extend the life of my bag, at least.
First, I loosened the screw slightly and worked the handle back into position. Then I tightened the screw up.
Step 2: Prepare the Sugru
Step 3: Apply the Putty
Then I pushed the handle down into the bag so that the Sugru wouldn't be in the way of the handle retracting once it has cured. It sticks up slightly higher because of the added material, but not so's you'd notice.
I left the Sugru to cure, and in the morning I had a usable rolling bag again!