All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
You mentioned some reprehensible matters about your father-in-law; if he really has done those things, then that was great evil. However, his misconduct and mistreatment does not exempt his children from being kind and dutiful to him, as they are obliged to be so with him in all cases. Being kind and dutiful to him is even confirmed as he has reached this old age.
Allah says (what means): {And your Lord has decreed that you not worship except Him, and to parents, good treatment. Whether one or both of them reach old age [while] with you, say not to them [so much as], "uff," and do not repel them but speak to them a noble word.} [Quran 17:23]
Besides, Abu Hurayrah, may Allah be pleased with him, narrated that the Prophet said, “Let him be humbled into dust; let him be humbled into dust; let him be humbled into dust.” It was said, O Messenger of Allah! Who is he?” He said, “The one who is with his parents during their old age, either or both of them, but he does not enter Paradise.” [Muslim]
The bad actions of the father do not justify his son to sever the ties of kinship with him or be negligent in fulfilling his obligations towards him. So your husband should fulfill his father's needs, whatever they are; spend on him; provide him with clothing and treatment; and so forth, if there is no one else among his children or others to do this for him.
Therefore, if the nursing home does not provide him with what he needs, then putting him there is wasting him and neglecting his right. So you should inform your husband about this and warn him of the bad consequences of severing the ties of kinship with his father. If he accepts and keeps his father with him, then this is what is required; otherwise, you are not Islamically obliged to serve your father-in-law or help him. So if your husband does not agree with that, then you should forget about the matter and be keen on preserving your family and your relationship with your husband and advise him in a gentle manner regarding any negligence towards his father. Please, refer to fatwas 87019and 82254.
On the other hand, what you mentioned about the eldest son abusing his father and expelling him from the house is a great evil, and this is being undutiful and disobedient to him, as it is evil to meet kindness with abuse. This son must be advised to repent to Allah and be kind and dutiful to his father. It should also be clarified to him that the inheritance money is not lawful for him, as it is not permissible for a father to favor one of his children over others when gifting them. It is an invalid gift unless his siblings approve of it, provided that they are all adults and of sound mind. For more benefit, please refer to fatwa 116350.
Allah knows best.