I saw a study that claimed improvement in sufferers of sleep apnea who played the didgeridoo.
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.