The rapid adoption of artificial intelligence (AI) in the public sector is reshaping how governments design policies, deliver services, and manage institutional knowledge. However, AI integration in digital governance often remains fragmented, with limited design-oriented frameworks that connect technological capabilities, knowledge management, and governance processes. This study addresses this gap by proposing an AI-driven knowledge governance framework for public sector digital transformation, explicitly aligned with system design principles and socio-technical architecture. Drawing on interdisciplinary perspectives from digital governance, knowledge management, information systems, and design research, the paper develops a structured framework incorporating data ecosystems, organizational memory, intelligent decision-support systems, and governance feedback loops. The study further presents design methods for integrating AI into public administration, including system architecture, workflow alignment, and visualized governance outputs that enhance transparency, accountability, and decision intelligence. Special attention is given to emerging economies, where institutional constraints, limited data infrastructure, and governance capacity gaps create unique implementation challenges. Ethical AI use, explainability, trust, and responsible automation are also examined as essential design requirements. By bridging AI innovation with governance design, the paper contributes a practical and scalable model for building inclusive, knowledge-driven, and resilient digital public sector systems.