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She insisted on helping me and wouldn't take no for an answer.
Literal
She [topic-は] me [object-を] will-help [quotative-と] saying didn't-listen.
The fixed pattern ~と言ってきかない (or ~と言って聞かない) — literally '[X], saying that, doesn't listen' — is the standard idiomatic frame for someone insisting on something and refusing to be talked out of it. きかない here is the negative of 聞く in its 'listen, heed' sense, not 'hear.' The pattern frames the insistence as stubbornness toward all counterargument. The plain 手伝う ('help') is in non-past form even within a past narrative because what's being reported is what she kept saying.