We present an algorithm for skill discovery from expert demonstrations. The algorithm first utilizes Large Language Models (LLMs) to propose an initial segmentation of the trajectories. Following that, a hierarchical variational inference framework incorporates the LLM-generated segmentation information to discover reusable skills by merging trajectory segments. To further control the trade-off between compression and reusability, we introduce a novel auxiliary objective based on the Minimum Description Length principle that helps guide this skill discovery process. We test our system on BabyAI, a grid world navigation environment, as well as ALFRED, a household simulation environment. Our results demonstrate that agents equipped with our method can discover skills that help accelerate learning and outperform baseline skill learning approaches on new long-horizon tasks.