So, Korean has many levels of formality, not just two. But the most common in every day speech is informal (with friends), casual (for strangers, colleagues and acquaintances of same age and hierarchy) and formal. Jaekyung speaks informally with Kim Dan throughout the conversation, but Kim Dan speaks casually until he says "No, I don't want to..." the first time (싫습니다...) at which point he starts speaking formally. The difference is clear-cut. Jaekyung flinches and scrunches his nose, not only at the reply, but also at the sudden level of formality. The message is clear, Kim Dan is putting distance between them.
Sorry, I haven't been reading this series in Korean. It comes out at the same day in English and Korean. I only went to check it this time because you asked. If I had to guess, based on his personality, I'd say Kim Dan was initially formal and then started speaking casually, but never informal. But I really don't know.
can someone who knows korean tell me if kim dan was talking formally or informally with jjk?