Answer: What you describe sounds like a disease of sorts, but I can't really determine which without inspecting the plant. Some of the most common problems with kalanchoe include Rhizopus wich can develop in warm, humid conditions. The fungus is an opportunistic pathogen that infects weak, soft tissues. Pythium is another disease.
The root system of Kalanchoe is very sensitive to inadequate watering. Too wet or too dry will cause problems. Kalanchoe can survive on very few roots. If the conditions for the roots are not perfect, the plant will simply abandon part of its root system which will allow Pythium infections to get established. A variety of stem rots can affect Kalanchoe (Phytophtora, Thielaviopsis, Rhizoctonia, Cylindrocarpon and others). The best course of action for you is to cut away the affected plant parts, including roots attached to the affected stems. This will remove the problem and also provide more room for the roots to grow. I would simply unpot the plant and cut away the plant parts, then repot in fresh potting soil. Best wishes with your kalanchoe.
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