잠들다 is the moment of dropping off, not sleeping in general — that is 자다. The -게 adverb describes how it happens: 행복하게 잠들었다 'fell asleep happy', 자연스럽게 잠들고 'just drifted off', 늦게 잠든 'fell asleep late', 나도 모르게 잠들고 'nodded off without realising'. Most of these lines are cosy, everyday talk about bedtime.