What should I do about my repeated discovery of my father's illicit relationship, knowing his illness and old age?

You must be dutiful to your father, and his right over you is not forfeited by his wrongdoing. You should speak frankly with your father and ask him about his relationship with that woman. If it is illicit, then you must gently and courteously forbid him from this evil, while covering up his fault. Enjoining good and forbidding evil for parents is not like doing so for others.