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She won't even speak to me.
Literal
She [topic-は] me [to-に] mouth [even-も] use-for-me-not.
口をきく (often written 口を利く) literally 'to use one's mouth,' meaning 'to speak to, to address.' Its negative form 口もきかない/きいてくれない is the standard idiom for 'not speak to (someone),' often as a sign of being upset or giving the silent treatment. Note the particle swap: を replaced with も ('even') intensifies — not 'doesn't speak' but 'doesn't even speak.' The negative auxiliary ~てくれない ('won't do for me') sharpens the speaker's grievance: she's not just silent, she's pointedly withholding the kindness of speech.