。
He vowed never to speak to her again.
Literal
Her [to-には] never-again will-not-speak-to [quotative-と] vowed.
Notable for the missing subject — the vow is made by some other party (likely 'he' or 'I' from context), not by 彼女. Japanese routinely omits subjects when context makes them clear. 彼女には marks her with には (に + contrastive は) as the addressee of speaking. 二度と...ない is the 'never again' pattern, here in the embedded vow.