Knowledge components (KCs) are key to assessing student knowledge levels on fine-grained skills and driving personalization and feedback. However, crafting KCs and tagging them for problems, traditionally performed by human domain experts, is highly labor-intensive. Prior work has studied automated KC generation only for multiple-choice questions but not open-ended ones. We bridge this gap and present an automated, large language model (LLM)-based pipeline for KC generation and tagging for open-ended programming problems. We also develop an LLM-based knowledge tracing (KT) framework to leverage these LLM-generated KCs. We conduct extensive quantitative and qualitative evaluations on two real-world student code submission datasets. Results show that our KT method outperforms existing ones and LLM-generated KCs outperform human-written KCs on future student response prediction. We also investigate how these KCs enable us to analyze student learning curves and conduct human evaluation with course instructors to further verify the quality of KC-problem tagging.