So Lyn Provost is now considering the request and hopes to make a decision within two weeks.
As Auditor-General, she certainly has the powers to mount an effective inquiry. But into what exactly? Public confidence has been dented by the huge privacy lapse. But that is a matter for the Privacy Commissioner, not the Auditor-General.
There are several separate strands to this messy business. Nick Smith's conflict of interest in writing a reference for his friend and National Party associate Bronwyn Pullar has nothing to do with ACC's big mistake. Neither does Smith's consequent departure from the Cabinet fall within Provost's purview.
Smith, however, wants an inquiry so he can prove his indiscretions were limited to some minor assistance for Pullar and that everything else he did while holding the ACC portfolio was above board.
Meanwhile, Pullar claims that her privacy was breached by her name being leaked from within ACC to the media.
It is a very serious allegation. But it follows years of argument and complaint by Pullar over the handling of her case by ACC after she was injured in a cycle accident.
Moreover, Provost's powers of inquiry do not extend to assessing the rights and wrongs of an individual claimant's case against the corporation.
More preferable would have been an inquiry conducted by a senior lawyer, such as a Queen's Counsel, who could have been given a wider brief and not been proscribed by statute as Provost is.
It is still possible something extraordinary may yet emerge which forces John Key to u-turn on his refusal to establish such an inquiry - one which would probe far wider than the joint inquiry being conducted by the Privacy Commissioner and ACC.
What has already happened is extraordinary, so such an eventuality cannot be ruled out.
An inquiry would be lose-lose for Key. If it uncovers something really dicey, that can only mean trouble for National. If an inquiry reported little or nothing was wrong it would be dismissed as a whitewash.
Smith's hopes of getting an inquiry were always forlorn. His resignation on Wednesday made him party to an unspoken deal with Key that he won't get one.
Smith has been a long-serving and loyal servant of the National Party. Having done the right thing by resigning, he can be assured the party will now look after him. But that means no inquiry. His seat at the Cabinet table will be filled by someone else. But Smith's intelligence, energy and salesmanship will not go to waste.
Once the current hoo-hah dies down, it is not inconceivable that in a few months he might be offered a ministerial post outside the Cabinet.
He is too valuable to allow to go to seed on National's backbenches for the best part of three years.
The dignified manner of his departure has already placed him on the road to rehabilitation.
That he somehow managed to hold himself together on Wednesday afternoon was something of a minor miracle.
His haggard face, sagging jowls and teary eyes were the look of someone who had not slept in days. It was the look of someone who had hit rock-bottom. It was the look of someone at the end of their tether - or close to it.
It took true courage to stand up in Parliament and announce his resignation as a Cabinet minister before stepping out of the chamber to face the waiting media.
Lesser politicians whose worlds had just crashed around them would have barged their way through the throng of microphones and cameras without stopping.
Smith - polite and professional to the end - endured the interrogation until the questions dried up and his colleagues could finally shepherd him away.
Jim Anderton liked to say that one bad day in Government was better than a thousand good days in Opposition.
But Wednesday was an especially bad day for National. Barely four months into its new term, the governing party has lost its first minister - and a senior one to boot.
A wheel has come off the National juggernaut. It is still full speed ahead with the party's reform agenda. But National has suffered its biggest psychological hit since Key became leader more than five years ago.
The sheer speed of Smith's demise was a shock for the party. On Monday afternoon, as the Local Government Minister, he stood alongside Key on the podium at the Prime Minister's weekly press conference to launch National's eight-point plan to curtail local body rates rises.
Less than 48 hours later Smith, who had also held the Environment and Climate Change Issues portfolios, was no longer the minister of anything.
There was astonishment and puzzlement within National that - despite all his experience - he should have ignored strict Cabinet rules on conflicts of interest to help Pullar in her fight against ACC when he was in charge of that portfolio at the time. It was not as if he was doing it for any apparent financial or personal gain.
Correspondence released this week shows Smith was perfectly well aware of the conflict of interest. His mistake was to think he could minimise it to a point where people would think it was not that big a deal in the grander context of his contribution as a hard-working minister and MP.
He might get into some minor trouble over it, but it would stop Pullar pestering him to intervene in her case.
Or so he seems to have thought. Smith could not have got things more wrong. But then Smith has often ridden his luck in the past. This time it ran out. Just as it seems to be running out for National generally.
