A flowering flapjack rosette may decline after its display. Look for healthy offsets at the base before deciding what part of the plant to remove.
The flowering rosette
Flapjacks such as Kalanchoe luciae are monocarpic: the rosette that flowers eventually dies. That is a different process from an otherwise healthy plant losing a few older leaves. Check the plant's identity before applying this rule to another succulent.
Look for offsets
A plant may produce offsets near its base or lower flower stalk. Check which parts remain healthy before discarding the container. An offset and the flowering parent are at different stages of growth, even when they still share a pot.
Keep root care sensible
Extra fertilizer does not reverse a flowering rosette's life cycle. Maintain suitable drainage and assess the remaining offsets on their own condition. Do not keep the whole pot constantly wet in an attempt to save an aging flowering stem.
Related reading: Flapjack Succulents Outdoors: Sun and Rain.