And if the plant is big enough, division beats cuttings in winter by a wide margin, because every piece already has roots. Out of the pot, find natural sections that each have leaves and their own roots or rhizome, separate by hand or with a clean knife, pot into dryish well-draining cactus mix, and wait about a week before watering so damaged roots can heal.
Divisions carry on growing with barely a pause even in October. The snapped leaf is a bonus experiment rather than your only option.