In The Scarlet Letter, why does Hawthorne say "the scarlet letter had not done its office" in chapter 13?

In addition to contributing to the growth of independence in Hester, rather than subjugation to the stringencies of the Puritan community, the scarlet letter has also "not done its office" in its failure to elicit from Hester Prynne her confession of the identity of the man with whom she has committed her sin of adultery.  For, in Chapter III, one of the townsmen remarks,

"Madame Hester absolutely refuseth to speak, and the magistrates have laid their heads together in vain."

The Reverend Mr. Wilson has exhorted her to indicate the man, telling her to speak his name that with her repentence, she "may avail to take the scarlet letter off thy breast."But, Hester adamantly replies, "Never!"

In Chapter XIII, then,the letter has most certainly "not done its office" as Scarlet wears the letter on her own terms, resolving to go completely counter to the wishes of the magistrates; for, she is determined to meet her former husband, Roger Chillingworth, and do whatever is in her power to rescue Dimmesdale, the "victim on whom he had so evidently set his gripe."

Chapter XIII of The Scarlet Letter, appropriately titled "Another View of Hester", is set seven years after the birth of Pearl. At this point,  Pearl is an older child but still challenging to Hester, Reverend Dimmesdale is showing the signs of decrepitude that are finishing slowly with his life, and Chillingworth is at his worst trying to probe Dimmesdale's mind.

Yet, Hester's presence demonstrates a strength and calmness of character that, as the narrator states,

It is remarkable, that persons who speculate the most boldly often conform with the most perfect quietude to the external regulations of society...So it seemed to be with Hester.

With these words the reader gets a perception of Hester that surprises even Hester, herself. No longer is she the child of passion and rebellion that led her to the affair with Dimmesdale. The passing years have awarded her with enough time to analyze her own behavior in depth. Even after the scaffold, the imprisonment, the public scorn, and her general humilliation, Hester Prynne still looks back and sees that, perhaps, her actions were not made with the purpose of sinning; she simply is a woman ahead of her times. She thinks differently, and feels differently.

She sees how her own strength of character and temperance far surpasses Dimmesdale's and Chillingworth's, even though both men are educated and more powerful, in comparison to her. Hence, Hester is slowly realizing that, in the end, the same letter which was first used to insult her, deter her, and limit her, has actually empowered her, and taught her how to become independent, and in peace with herself.

She might have come down to us in history, hand in hand with Ann Hutchinson, as the foundress of a religious sect. She might, in one of her phases, have been a prophetess.

Therefore, the original purpose of the scarlet letter has never been met. Far from submitting Hester to the rules of the settlement, the letter effectively detached Hester from her insignificant and ignorant surroundings. The letter taught her how to appropriately measure the influence of other people's thoughts, beliefs, and opinions, on her life. Hester basically dominated and transformed the scarlet letter; it did not happen the other way around, as was expected.

[It] had not done its office.

What does the line "the scarlet letter has not done its office" mean?

This phrase comes from Chapter XIII of The Scarlet Letter, a chapter that many have perceived Hester as, like Ann Hutchinson, a rising feminist.  For, while she has subjugated her independence of thought to the care and safety of her child, she has not achieved repentance for her sin with a renewal of Puritan faith.  Instead, she feels a cold despair, wondering if

it were for ill or good that the poor little creaature had been born at all.

In fact, the same question arises in Hester's mind regarding "the whole race of womanhood." She ponders the "hopeless task" of independence for women who must abandon the priorities of the heart if they are to be able to achieve any recognition as individuals and who must see an entire reform of society if there are to be "mightier reforms."  Certainly, they cannot be a part of the Puritan community with its denial of the spirit of woman.  Ironically, then, it is Hester's mark of the scarlet letter and isolation from the Puritan community that has drawn her farther away from the thought of her sect as it has afforded her the distance and independence necessary to such ponderings.  Indeed, the scarlet letter has not done its office of molding Hester into a true Puritan.