Event cameras offer high-temporal-resolution sensing that remains reliable under high-speed motion and challenging lighting, making them promising for localization from LiDAR point clouds in GPS-denied and visually degraded environments. However, aligning sparse, asynchronous events with dense LiDAR maps is fundamentally ill-posed, as direct correspondence estimation suffers from modality gaps. We propose LEAR, a dual-task learning framework that jointly estimates edge structures and dense event-depth flow fields to bridge the sensing-modality divide. Instead of treating edges as a post-hoc aid, LEAR couples them with flow estimation through a cross-modal fusion mechanism that injects modality-invariant geometric cues into the motion representation, and an iterative refinement strategy that enforces mutual consistency between the two tasks over multiple update steps. This synergy produces edge-aware, depth-aligned flow fields that enable more robust and accurate pose recovery via Perspective-n-Point (PnP) solvers. On several popular and challenging datasets, LEAR achieves superior performance over the best prior method. The source code, trained models, and demo videos are made publicly available online.