。
She refused to let me read the letter.
Literal
She [topic-は] me [to-に] that letter [object-を] cause-to-read [for me-てくれよう] [would not try-としなかった].
A grammatical layer cake. (1) 読ませる is the causative of 読む — 'cause/let to read.' (2) ~てくれる is the benefactive auxiliary — 'do for me.' Together, 読ませてくれる = 'let me read (as a favor).' (3) Volitional ~よう then conjugates 読ませてくれる into 読ませてくれよう ('let's let me read'). (4) ~ようとしない negates the volitional + とする ('try to') frame: 'won't try to / refuses to.' The whole stack: 'refused to even consider letting me read.' One of those classic Japanese sentences where Western learners begin to appreciate how aspectual layers stack into precise meaning.