No way out, but my mental health is terrible. Any advice?

I think there is a lot of denial here about the circle of life if you are unable to accept that your beloved Dad is nearing the end of his life, and if you are investing yourself to this extent in your father and only your father. I think surely you understand that this isn't the normal cycle. Our parents raise us to be independent and to have our own lives, while their lives fade. It would break my heart to believe my daughters didn't accept now that they will soon in fact lose me (I am 79). I have already long discussed with them that they are not to take me into their care, that I will enter care of ALF, then Nursing Home. They are grown. They have their lives. It is what I raised them to do. They are raising the next generation. That's their job. My eldest is 59. Is she to give up some of the most quality years of her life now, her own child raised, I would jump off a bridge first.
If you already fear so to lose your Dad, then putting him into care in his last year or so would likely lead you to inappropriate and devastating guilt. You are building you life on the generation that is gone now, not on the future for our world.
You say that your mental health is in the gutter. There is more going here than your caregiving of your Dad, and I do hope very much that you are seeking the BEST professional help to sort all this out and to help you. Your Father's other daughters are taking the path toward the future, as is the grandchildren.
In the end, and after exploring choices and feelings and beliefs, you choose to care for your father until his death then that is your honest and real choice. Knowing fully that it is your choice should help you. And knowing that you have done all you chose to do should lessen your guilt. I have not only long ago lost my parents, but more recently the brother who was Hansel to my Gretel in every forest of life. I will tell you that they really never leave you. They are always with you in the wonder of the relationship you shared through life.
I will leave you with my favorite Annie Dillard quote: "We live our lives as though hundreds of thousands of generations had not come before us, and as though there were not hundreds of thousands of generations still to come". Our lives are as nothing. A blip in time. But for me, it is imperative to use the each to live with as much quality of time, joy, and care for our earth as we are able to.

((Hugs)) for you! It's really tough to be alone in this battle. But I hope you take time to care for yourself. Please get help where you can. I know that not everyone is willing or can help, but perhaps you can reach out to organizations or hired aides in your area. Try to think of certain specific help that you need and build that support system bit by bit.
I, too, care for my father alone, with my sibling rarely involved. It helped a lot to have someone come in to clean the house, to have meals delivered, etc. because it took some load off my back. Not everything can be solved, but at least some breathing space became available.
Then maybe you can have the space to think clearly through some long-term options as well. Hang in there! Praying for you.

Your father would not want you suffering because of him. You are now at the point where you must so what is best for you and your mental health. Your father has lived his life and it's time that you live yours too.
It sounds like your father is perhaps depressed, and placing him in the appropriate facility might actually be helpful for him as it would give him other folks his age to interact with. You might be pleasantly surprised how well he might do in such a place. You say that you have no way out, but you do have options here. You have to first and foremost take care of yourself, or there is no way you can be of any good to him. Please start making yourself a priority.

Hi Sun, I think it is time to make a decision ASAP. First of all your health is at risk. What would happen to your dad if you were to get ill. I would place him in a decent facility near you or get home care. No other options in my opinion. Counseling might be beneficial for you. Sending my thoughts and prayers to you and your father.