How can we reconcile the Hadith of Anas (may Allah be pleased with him): "Do not say 'madman,' but rather say 'afflicted.' A madman is one who persists in disobeying Allah," with the Hadith: "And concerning the madman until he recovers his sanity"?

The first Hadith (definition), which describes one who persists in sin as insane, is not a denial of actual insanity. Rather, it is a rhetorical device to draw attention to the grave danger of persisting in sin, as the one who commits it acts like an insane person due to their disregard for the consequences. As for the second Hadith, it confirms that the pen is lifted from the insane, thereby removing sin and accountability from them. There is no contradiction between the two Hadiths.