What should I do about my father's pursuit of adultery at home, taking into consideration my inability to stop him, and his need for my care due to his illness and old age?

The Sharia (definition) has enjoined honoring one's parents and treating them kindly, and associating with them in a good manner, even if they are polytheists. These rights are not abrogated by any major sin short of polytheism. Obedience to Allah takes precedence over everything else; so if Allah's command conflicts with the parents' command, Allah's command is given precedence. Do not live with your father if he commits the abomination of adultery, out of fear for your religion. Continue to advise him and remind him of Allah through various methods and means, and be keen to search for a righteous wife for him who will make him chaste.