Does a divorce issued by a sick father who suffers from memory loss, mood swings, and blaspheming religion count as a valid divorce, and is it permissible to admit him to a nursing home?

If the father reaches a stage of dementia, religious obligations are lifted from him, and he is not held accountable for his actions. One can endure his actions or place him in a nursing home while continuing to show him filial piety, visiting him, and fulfilling his needs. Allah has enjoined kindness to parents, especially in old age, and prohibited showing annoyance or harming them verbally or physically. He commanded kind speech, humility, and praying for their mercy, as stated in His saying, the Almighty: (And your Lord has decreed that you worship none but Him. And to parents, show goodness. If one of them or both of them attain old age in your life, say not to them a word of disrespect, nor repel them, but address them in terms of honor. And lower to them the wing of humility out of mercy and say, "My Lord, have mercy upon them as they brought me up when I was small.")