cat with mange?

Question
11 year old neutered male cat. Starting getting black specs on his skin. chin and face mostly. He has lost over 3 lbs in the last month. Vet thought allergy, steriod made it worse. He is now pretty much completely bald on his legs and feet and the underside of his tail except for black scabby patches. His face is mostly bald, especially around his eyes, chin and down his chest. He is licking and biting to try to get the black off. We are thinking it is feline mange, which is rare, but possible. He is being treated with Ivermectin injections but our vet seems very unfamiliar. From what I have read he should be getting injections every 7-10 days. She is insisting on every 2-3 weeks. After the first injection, he seemed to perk up and his skin started clearing up, but by the time he got his second injection, his skin was getting worse again and he was pretty lethargic. He has also refused to use his litterbox and will go to the bathroom on the floor right in front of us! He is quite sick, and has a hard time even walking, his feet hurt him a lot. He shakes his head, shakes his legs, twitches almost constantly. Any help with Ivermectin dosage would be appreciated. Reinforcement of the mange diagnosis would be appreciated.
We are getting desperate and considering putting him down because he seems so miserable. But we really don't want to do this.

Answer
For mange in cats, they need to be dipped in LymSulfur.  Ivermectin will help only a few mites.  I would suggest a diet change to a hypoallergenic diet, like zd for 6 weeks to see if it is a food allergy and if that is not helping, then I would biopsy several of these black comedones to see if there are mites or fungi in the skin.