I haven't found a new ball of the right size and make yet though, so last night I decided he might as well try Lada's old ball.
It wasn't the greatest success at first. He basically sat down and ate his broccoli. And this was hilarious: apparently he had to go to the bathroom. So there were a few pieces of poop in his ball. He kept trying to move those away from where he was sitting, sort of pushing them out of the way. But he was in the ball, so the floor wasn't flat and the poop kept falling back towards him!
Anyway, he had a run around the guest bathroom and I removed the poop from the ball. When he got back in the ball after that, he had a decent run around the living room.
Last night's video evidence: