I know he has the best voice therapists in the world and there's no chance he'll ever read this but: kazoo. That's my advice for anyone who has to give a speech or get their voice in shape for any reason. Ukulele & a kazoo 45 minutes every day.
Not sure why you think humming/playing kazoo should be a good exercise for someone who has damage in the vocal cords. E.g. the common medical advice for someone who with a hoarse throat (due to a cold etc) is to let it rest, i.e., refrain from speaking. But I'd be happy to see studies proving that wrong.