Traditional automated machine learning (AutoML) often faces limitations in manual effort, complexity management, and subjective design choices. This paper introduces a novel LLM-driven AutoML framework centered on the innovation of decomposed prompting. We hypothesize that by strategically breaking down complex AutoML tasks into sequential, guided sub-prompts, Large Language Models (LLMs) operating within a code sandbox on standard PCs can autonomously design, implement, evaluate, and select high-performing machine learning models. To validate this, we primarily applied our decomposed prompting approach to sleep disorder classification (illustrating potential benefits in healthcare). To assess the generalizability and robustness of our method across different data types, we subsequently evaluated it on the established 20 Newsgroups text classification benchmark. We rigorously compared decomposed prompting against zero-shot and few-shot prompting strategies, as well as a manually engineered baseline. Our results demonstrate that decomposed prompting significantly outperforms these alternatives. Our results demonstrate that decomposed prompting significantly outperforms alternatives, enabling the LLM to autonomously achieve superior classifier design and performance, particularly showing strong results in the primary sleep disorder domain and demonstrating robustness in the benchmark task. These findings underscore the transformative potential of decomposed prompting as a key technique for advancing LLM-driven AutoML across diverse application areas beyond the specific examples explored here, paving the way for more automated and accessible problem-solving in scientific and engineering disciplines.