My Dad wants to die but doesn't have a life-threatening diagnosis. How can I determine if Hospice is right for him?

Both parents alive, Mom is 85, Dad is 87. They live alone in a single family home. Dad's health is bad requiring 24/7 care. Severe stroke, paralyzed on right side, and memory probs but NO diagnosis of Alzheimers (he knows all of us, our names etc) He can't walk anymore....not without a lot of assistance. Mom is in denial about his condition. Dad says he wants to die. Severe diabetes requiring insulin, difficulty eating, swallowing, following conversations, severe mood swings, tried to kill himself by taking a bunch of Aleve pills (the only pills he could find) but couldn't manipulate the pills well enough to take them, spilled all over the floor instead. Is definitely suffering due to not possessing all his faculties or being able to walk etc. Mom doesnt want to do Nursing Home (altho that is the obvious need here) because she 'cant live without him' and he doesn't want to leave home. Money for a fulltime nurse is becoming impossible. Am thinking about hospice care instead...especially since HE has expressed a desire to die. But NO firm diagnosis of a death sentence in 6 months or whatever form a doctor. SOME Alzheimers symptoms but definitely not all. He is emotionally abusive and has the maturity of a 4-year-old. I don't know what to do. Mom is struggling, trying to care for him with help from a nurse several x a week and friends and family, but it's becoming too much. Falls are common; and Mom's balance isn't great either so they have BOTH fallen! Bottom line: Dad needs 24/7 care IN HIS OWN HOME. I know hospice could give that to him but Mom won't admit he is at that point yet.
I need more info on hospice and the conditions/diseases which qualify. Assisted living will NOT take Dad....too many medical issues. Nursing home is probably best but they will not separate and Dad wants to die at home.
HELP!