。
She entertained the child until they laughed.
Literal
She [topic-は] child [object-を] coaxed-and-entertained, made-laugh.
あやす is a beautifully specific verb — 'to coax, soothe, entertain a baby or small child' through funny faces, gentle bouncing, peekaboo, and whatever else stops the crying or sparks a giggle. It pairs perfectly with the causative 笑わせる ('made laugh'), with the te-form of あやす chaining into the resulting action. Two verbs, one tender little scene — an example of how Japanese can pack vivid action sequences into compact sentences.