You must be made of steel to cope so well with this situation. I don't think I could do it. Like others here I'm becoming concerned with your health, both mental and physical.
Surely if you discussed this with Mike he would agree to go into an AL situation for awhile. At least until his sister was stabilized or perhaps you can hire (I know how you feel about this) someone to come in for night hours to care for Mike while YOU got some rest, some sleep. Here these people cost $18 to $20 hr with a minimum of 3 or 4 hours at a time. Or you can pay them for the 8 hrs of a night. They would be no bother to the rest of the household since they will sit with him and not the rest of the family.