To an extent, the visit must be judged a success. Mr Annan has put his reputation on the line promising to propose a comprehensive package of UN reforms by the summer. He has also fully engaged Washington on its responsibility to deliver the other side of the bargain: to pay up its back dues to the UN, worth over $1.3bn, and thus relieve the organisation's financial crisis.
That this is going to be a hard trick to pull off is evident. But as Mr Annan flew back to headquarters last Friday he must have pondered that something worse than failure might happen - that the demands coming from Capitol Hill may ignite months of dangerous argument between the United States on the one hand and the rest of the UN on the other, with him, all reasonableness, in the middle.
For example: Mr Annan suggested during a speech to the National Press Club in Washington last Friday that the US might be allowed to lower its long-standing 25-per cent share of the regular UN budget, if it could persuade other member states to acquiesce. "The formula is not sacrosanct," he intoned. "The United States can negotiate a lower rate for itself". That came only a day after Mr Helms, dawdling with reporters outside the White House, had mused about those "European countries that are so critical about us". Perhaps, he said, it was time "we looked at what they contribute".
Mr Annan conceded another point to Mr Helms. The senator is proposing to table draft legislation on the Hill as early as this week laying down what he called "benchmarks" of reform that the UN will have to meet before Congress will consider relinquishing monies to pay back its UN arrears. And Mr Annan has agreed to receive staff members from the senator's office in New York this week to discuss what the legislation might look like.
Defensive UN officials insisted that Mr Annan has little choice but to show accommodation to the US. Mr Clinton, they said, urged Mr Annan to make friends with as many people as possible on Capitol Hill. After all, it is the members of Congress who are refusing to pay the UN. The administration is fully aware of the damage being done to its influence in the UN.
Mr Annan's overtures are certain, however, to make some of the 184 UN members nervous, if not livid. The US, after all, does have a treaty obligation to pay its dues on time. Why, other countries will wonder, the special treatment for Washington?
Mr Annan's agreement to see staff from Senator Helm's office came only hours after he told an editorial board of the Washington Post that he would not deal directly with officials from member state parliaments, only with national missions established in New York, he said.
The idea of "benchmarks" will not sit well with any other state, Britain and France among them. For benchmarks read conditions - conditions on fulfilling a treaty requirment. Questioned in a corridor of the Senate last Thursday, a UN staff member described the Helms demands as being "sticky" for Mr Annan and that he was going to have to walk "a fine line". Indeed.