Communication infrastructure in remote areas struggles to deliver stable, high-quality services for power systems. Low Earth Orbit (LEO) satellite networks offer an effective solution through their low latency and extensive coverage. Nevertheless, the high orbital velocity of LEO satellites combined with massive user access frequently leads to signaling congestion and degradation of service quality. To address these challenges, this paper proposes a LEO satellite handover strategy based on Quality of Service (QoS)-constrained K-Means clustering and Deep Q-Network (DQN) learning. The proposed framework first partitions users into groups via the K-Means algorithm and then imposes an intra-group QoS fairness constraint to refine clustering and designate a cluster head for each group. These cluster heads act as proxies that execute unified DQN-driven handover decisions on behalf of all group members, thereby enabling coordinated multi-user handover. Simulation results demonstrate that, compared with conventional handover schemes, the proposed strategy achieves an optimal balance between performance and signaling overhead, significantly enhances system scalability while ensuring long-term QoS gains, and provides an efficient solution for mobility management in future large-scale LEO satellite networks.