Ensuring accessibility in mobile applications remains a significant challenge, particularly for visually impaired users who rely on screen readers. User interface icons are essential for navigation and interaction and often lack meaningful alt-text, creating barriers to effective use. Traditional deep learning approaches for generating alt-text require extensive datasets and struggle with the diversity and imbalance of icon types. More recent Vision Language Models (VLMs) require complete UI screens, which can be impractical during the iterative phases of app development. To address these issues, we introduce a novel method using Large Language Models (LLMs) to autonomously generate informative alt-text for mobile UI icons with partial UI data. By incorporating icon context, that include class, resource ID, bounds, OCR-detected text, and contextual information from parent and sibling nodes, we fine-tune an off-the-shelf LLM on a small dataset of approximately 1.4k icons, yielding IconDesc. In an empirical evaluation and a user study IconDesc demonstrates significant improvements in generating relevant alt-text. This ability makes IconDesc an invaluable tool for developers, aiding in the rapid iteration and enhancement of UI accessibility.