The overhaul will require big banks to put more money aside against future losses to curb excessive risk-taking.
"No one was responsible for protecting the whole system from the kind of risk that tied these firms from one another," President Barack Obama said.
"Regulators were charged with seeing the trees, but not the forest. We were facing one of the largest financial crisis in history and those responsible for oversight were most caught off-guard and without the authority needed to address the problem."
In the UK, oversight of the financial system is carried out jointly by the Treasury, Bank of England and city watchdog, the Financial Services Authority.
But Mr King, delivering the annual Mansion House speech, said the major lesson of the financial crisis was that these authorities lacked the means to take effective action to prevent excessive risk taking at banks.
He said reform of the Bank's powers was vital if it was to fulfil the role it had been asked to.
"Warnings are unlikely to be effective when people are being asked to change behaviour which seems to them to be highly profitable," he said.
The Treasury has resisted significant changes to the allocation of regulatory responsibilities
"So it's not entirely clear how the Bank will be able to discharge its new statutory responsibility if we can do no more than issue sermons or organise burials.
"Blaming individuals is no substitute for acknowledging the failure of the system."
The chancellor, on the other hand, did blame those at the heads of financial institutions.
And while he said there could be no return to "business as usual" for UK banks, he did not lay out plans for radical, structural change.
Instead, he called for "a change of culture" in which bank staff were "rewarded for long-term success, not for failure".
"Bank boards must have the right people of the right skills and the right experience to manage themselves more effectively," the chancellor said. "And they need to be equipped to ask the right questions.
"Their focus must be on long-term wealth creation and not short-term profits."
Mr Darling's approach came in contrast to plans unveiled earlier on Wednesday by US President Barack Obama, which involved the most significant reform of finance regulation since the 1930s Great Depression.
Alistair Darling: "We cannot go back to business as usual"
The BBC's business editor Robert Peston said the comments by Mr King were bound to be seen as an attack on the chancellor because "the Treasury has resisted significant changes to the allocation of regulatory responsibilities between the Financial Services Authority, the Treasury and Bank of England".
Mr King also outlined what he saw as the problems of the finance sector more widely, saying it had become too "big and too highly leveraged".
"One important practical step would be to require any regulated bank itself to produce a plan for an orderly wind down of its activities," he said.
Banks had entered the financial crisis with "inadequate" levels of capital to absorb losses, he said, suggesting that smaller institutions might be a safer bet in the future.
He renewed his call for High Street banks to be prevented from straying into investment banking where they could take big risks while, at the same time the government provided "an implicit guarantee against failure".
"If some banks are thought to be too big to fail, then, in the words of a distinguished American economist, they are too big," he added.
Whisper it softly, but there was the beginning of a grown-up debate about financial sector reform and public spending
But Angela Knight, of the British Bankers' Association, said that the "overwhelming majority" of UK banks had ridden out the downturn without needing direct bail-outs, and said the government should not look to restrict banks' business models.
"If we intervene and start to break up banks artificially, then the business goes elsewhere," she said.
Both Mr King and the chancellor spoke of signs of recovery in the economy but remained cautious.
Mr King said that "there are some signs that the British economy is beginning to stabilise", but said recovery could be "protracted".
The chairman of the Treasury Committee, John McFall, told the BBC said that the governor had been "sending a message to banks and financial institutions that it was not business as usual".
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