Which word describes someone who is stubbornly impenitent and hardhearted?

The term that best fits the description of someone who is stubbornly impenitent and hardhearted is "obdurate." This word specifically conveys a sense of inflexibility and resistance to changing one's mind or feelings, particularly regarding moral issues or personal behavior. In the context of impenitence, it denotes a person who refuses to acknowledge or feel remorse for their actions, exhibiting a hardened attitude.

While the other options may share some thematic elements, they do not denote the same degree of emotional rigidity or moral stubbornness. "Relentless" implies a continuous pursuit or insistence but does not specifically relate to a hardhearted nature. "Firm" suggests a strong stance but lacks the connotation of emotional or moral hardness inherent in "obdurate." "Recalcitrant," while it also suggests resistance to authority or control, does not carry the same connotation of emotional impenitence and hardness toward moral considerations.

Therefore, the distinct characteristics of "obdurate" make it the most accurate choice in this scenario.