Entity Linking (EL), the task of mapping textual entity mentions to their corresponding entries in knowledge bases, constitutes a fundamental component of natural language understanding. Recent advancements in Large Language Models (LLMs) have demonstrated remarkable potential for enhancing EL performance. Prior research has leveraged LLMs to improve entity disambiguation and input representation, yielding significant gains in accuracy and robustness. However, these approaches typically apply LLMs to isolated stages of the EL task, failing to fully integrate their capabilities throughout the entire process. In this work, we introduce DeepEL, a comprehensive framework that incorporates LLMs into every stage of the entity linking task. Furthermore, we identify that disambiguating entities in isolation is insufficient for optimal performance. To address this limitation, we propose a novel self-validation mechanism that utilizes global contextual information, enabling LLMs to rectify their own predictions and better recognize cohesive relationships among entities within the same sentence. Extensive empirical evaluation across ten benchmark datasets demonstrates that DeepEL substantially outperforms existing state-of-the-art methods, achieving an average improvement of 2.6\% in overall F1 score and a remarkable 4% gain on out-of-domain datasets. These results underscore the efficacy of deep LLM integration in advancing the state-of-the-art in entity linking.