The convergence of healthcare and financial data infrastructures driven by integrated billing systems, electronic health records (EHR), and digital payment gateways has created unprecedented attack surfaces for adversarial actors. Existing cyber-defense frameworks are predominantly domain-specific, addressing either clinical data protection or financial fraud prevention in isolation, yet failing to address the systemic interdependencies inherent in converged environments. This paper proposes a novel Self-Healing Cyber-Defense Framework (SHCDF) for converged healthcare-finance data infrastructures, drawing upon principles from self-healing networks, cyber-physical systems (CPS) resilience , and Large Language Model (LLM)-powered threat analysis. The framework integrates ML-based anomaly detection, Natural Language Processing (NLP) for cyber-threat intelligence extraction, and an LLM-driven reasoning engine that provides human-interpretable explanations for automated defense decisions. Evaluation across five threat categories ransomware, data exfiltration, phishing, payment fraud, and IoMT device compromise demonstrates an average recall of 89.6% and mean time-to-recovery (MTTR) of 3.3 minutes. Our findings contribute a generalizable architecture for cross-domain cyber resilience and highlight open challenges in regulatory compliance, model interpretability, and deployment scalability.