Artificial Intelligence is rapidly advancing and radically impacting everyday life, driven by the increasing availability of computing power. Despite this trend, the adoption of AI in real-world healthcare is still limited. One of the main reasons is the trustworthiness of AI models and the potential hesitation of domain experts with model predictions. Explainable Artificial Intelligence (XAI) techniques aim to address these issues. However, explainability can mean different things to people with different backgrounds, expertise, and goals. To address the target audience with diverse needs, we develop storytelling XAI. In this research, we have developed an approach that combines multi-task distillation with interpretability techniques to enable audience-centric explainability. Using multi-task distillation allows the model to exploit the relationships between tasks, potentially improving interpretability as each task supports the other leading to an enhanced interpretability from the perspective of a domain expert. The distillation process allows us to extend this research to large deep models that are highly complex. We focus on both model-agnostic and model-specific methods of interpretability, supported by textual justification of the results in healthcare through our use case. Our methods increase the trust of both the domain experts and the machine learning experts to enable a responsible AI.