Large Language Models (LLMs) offer many opportunities for scalably improving the teaching and learning process, for example, by simulating students for teacher training or lesson preparation. However, design requirements for building high-fidelity LLM-based simulations are poorly understood. This study aims to address this gap from the perspective of key stakeholders—teachers who have tutored LLM-simulated students. We use a mixed-method approach and conduct semi-structured interviews with these teachers, grounding our interview design and analysis in the Community of Inquiry and Scaffolding frameworks. Our findings indicate several challenges in LLM-simulated students, including authenticity, high language complexity, lack of emotions, unnatural attentiveness, and logical inconsistency. We end by categorizing four types of real-world student behaviors and provide guidelines for the design and development of LLM-based student simulations. These include introducing diverse personalities, modeling knowledge building, and promoting questions.