In recent years, Large Language Models (LLMs) have demonstrated remarkable capabilities in understanding and solving mathematical problems, leading to advancements in various fields. We propose an LLM-embodied path planning framework for mobile agents, focusing on solving high-level coverage path planning issues and low-level control. Our proposed multi-layer architecture uses prompted LLMs in the path planning phase and integrates them with the mobile agents' low-level actuators. To evaluate the performance of various LLMs, we propose a coverage-weighted path planning metric to assess the performance of the embodied models. Our experiments show that the proposed framework improves LLMs' spatial inference abilities. We demonstrate that the proposed multi-layer framework significantly enhances the efficiency and accuracy of these tasks by leveraging the natural language understanding and generative capabilities of LLMs. Our experiments show that this framework can improve LLMs' 2D plane reasoning abilities and complete coverage path planning tasks. We also tested three LLM kernels: gpt-4o, gemini-1.5-flash, and claude-3.5-sonnet. The experimental results show that claude-3.5 can complete the coverage planning task in different scenarios, and its indicators are better than those of the other models.