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She insisted to me that she had done no such thing.
Literal
She [topic-は] never that-kind-of-thing [contrast-は] is-not-doing [quotative-と] me [target-に] insisted.
主張する ('to insist, claim, assert') describes the act of pressing one's position firmly — stronger than the neutral 言う ('say') or the neutral 述べる ('state'). The embedded clause uses ~ている in its resulting-state reading: そんなことはしていない ('hasn't done such a thing,' i.e. is in the state of not having done it). The contrastive は after こと marks 'such things [as opposed to whatever I might have done]' — a small particle that adds defensive specificity. The whole sentence has the flavor of a denial under accusation.