。
She finally calmed down once she stopped crying.
Literal
She [topic-は] crying-stopped [then-て] at-last calmed-down.
泣きやむ is another compound verb: 泣く ('cry') plus やむ (intransitive 'to cease, stop'). やむ specifically describes weather or physical phenomena coming to an end (雨がやむ 'the rain stops,' 風がやむ 'the wind dies down') and metaphorically applies to crying as a kind of natural occurrence that runs its course. The te-form 泣きやんで chains into the next clause as 'and then,' and ようやく ('at last, finally') signals that the calm came after some wait. 落ち着く ('to settle down, calm down') is a verb of intransitive change — its pair 落ち着かせる is the transitive 'calm someone else down.' The polite ました ending lifts the register to neutral-polite.