A weakness or disability, especially due to old age, is called an infirmity.
The noun infirmity, when used alone, is generally understood to mean physical frailty. If you want to imply mental weakness or some other disability, then use an adjective for clarity. His "mental infirmity" was caused by early onset Alzheimer's disease. Don't confuse infirmity with infirmary. The latter is a place where the sick and infirm go to recover: "Because of her infirmity, she often ended up being seen by doctors at the infirmary."
