As Large Language Models (LLMs) become increasingly embedded in intelligent tutoring systems (ITSs), a growing need exists to understand how learners engage with these tools, especially in cognitively demanding domains such as computer programming. While prior research has focused mainly on LLM-generated scaffolding, less attention has been paid to student-side engagement, including how learners think, respond, and regulate their learning during natural language tutoring sessions. This study addresses that gap by identifying learner engagement state profiles based on behavioral and metacognitive patterns in student dialogue. Drawing on data from 36 recorded LLM-mediated C introductory programming tutorials, 1,046 dual-annotated student utterances were analyzed using K-means clustering. The analysis revealed three distinct learner engagement state profiles: Passive Reactors, Clarification Seekers, and Reflective Performers. These engagement states differed in response accuracy, metacognitive expression, and interaction style. Passive Reactors showed low initiative and limited self-regulation; Clarification Seekers demonstrated moderate accuracy and reactive help-seeking; Reflective Performers exhibited strategic engagement and high metacognitive activity. This study introduces an exploratory, scalable approach to learner profiling through natural language dialogue, advancing the design of adaptive, learner-aware LLM tutoring systems. The findings support the development of real-time learner modeling techniques that move beyond correctness, offering actionable insights for delivering more personalized and effective AI-assisted instruction in programming education.