Their father is senile and divorced their mother – did divorce really take place? And should they put him in a home for the elderly?

It seems from what your father is doing that he has reached
the stage of senility, at which the obligations of sharee’ah are waived, so
he is not expected to pray or fast, and any vows, oaths or words of divorce
spoken by him are invalid. 

If you can be patient and put up with the things he does,
then you should do that. If you cannot bear it, then there is nothing wrong
with you taking him to a home that takes care of the elderly, on condition
that you carry on honouring him and visiting him, and meeting his physical
and emotional needs as much as you can. 

You should note that Allaah has enjoined honouring parents,
especially when they reach old age, because of their great need at that
time. Allaah has forbidden mistreating parents in word or deed, even by
saying “Uff” (a mild expression of displeasure). 

Allaah says (interpretation of the meaning): 

“And your Lord has
decreed that you worship none but Him. And that you be dutiful to your
parents. If one of them or both of them attain old age in your life, say not
to them a word of disrespect, nor shout at them but address them in terms of
honour

And lower unto them the
wing of submission and humility through mercy, and say: ‘My Lord! Bestow on
them Your Mercy as they did bring me up when I was young.’”

[al-Isra’ 17:23]

Shaykh ‘Abd al-Rahmaan al-Sa’di said: 

Then after mentioning His rights, Allaah mentions the rights
of parents and says, “And that you be dutiful to your parents” i.e.,
you should treat them kindly in all ways, in word and deed, because they are
the reason why a person exists and because they love their child and treat
him kindly, which confirms that the child in turn is obliged to treat them
kindly.  

“If one of them or both of them attain old age in your
life” means, if they reach the age at which they grow weak and need
kindness and good treatment, “say not to them a word of disrespect”:
the word “uff” is the least kind of offensive speech, and what is meant is
do not mistreat them in the slightest. 

“Nor shout at them” means, do not rebuke them or speak
to them in a disrespectful manner. “But address them in terms of honour”
means, in a polite, gentle and kind manner, which will soften their hearts
and give them peace of mind. That varies according to circumstances, customs
and times. 

“And lower unto them the wing of submission and humility
through mercy” means, be humble towards them and compassionate, seeking
reward thereby, not because you are afraid of them or hope for what they
have and other motives for which a person will not be rewarded. 

“And say: ‘My Lord! Bestow on them Your Mercy” means:
pray for Allaah’s mercy for them during their lifetime and after they have
died, in return for their having brought you up when you were young. 

From this it may be understood that the more time spent in
looking after their child, the greater their rights. Also, whoever takes
care of raising a person and teaching him properly about his religious and
worldly affairs instead of his parents has similar rights over the one whom
he raised. 

Tafseer al-Sa’di, p. 407, 408 

And Allaah knows best.