Wow, what a sentence! Are you reading Victorian novels?
I see more than one way to interpret this sentence. It's meaning might become clear if I were willing to read the whole novel (which I am not). This is my guess, which I do not guarantee to be correct:
There are 3 people mentioned in this sentence. Let's call them
AA - the penitent
BB - the female who greets AA
CC - Mrs Lynde
AA was not a person that was just a little bit sorry (a meek penitent). Rather AA is a genuinely penitent person. In fact, AA is so very penitent that BB feels motivated (it behooved her) to allow her to see CC.