。
She was sobbing so hard she couldn't speak.
Literal
She [topic-は] sobbed [so-て], mouth [subject-が] was-not-able-to-function.
泣きじゃくる is a specific verb for sobbing — crying convulsively, in racking sobs, the way a child does when overwhelmed. It's a single dictionary word rather than a productive compound: the じゃくる element doesn't appear independently in modern Japanese. The te-form 泣きじゃくって drives the cause-effect chain into 口がきけない — the same 'mouth-doesn't-function' idiom that marks any kind of overwhelming emotion, here choking on tears rather than freezing in fright.