。
She broke her promise to write to me.
Literal
She [topic-は] me [to-に] correspondence [object-を] give [the-promise-to-という約束を] didn't-keep.
An elegantly compact construction: ~という約束 ('the promise that ~') is a noun phrase formed by quoting the content of the promise and tagging it with という — like English 'the promise that.' The full phrase 私に便りをくれるという約束 ('the promise that [she'd] write to me') is the object of 守らなかった ('didn't keep'). 便り ('news, correspondence') is a slightly literary noun for written correspondence — older and more poetic than 手紙. 約束を守る ('keep a promise') is the standard idiom; the negation 守らなかった makes the breach explicit.