The Hirshhorn, one of the 19 museums of the Smithsonian Institution, has been in very serious, very public crisis. Through resignations both forced and unforced over the past 18 months or so, the Hirshhorn has lost its chairwoman, its director, its chief curator and half its other board members. In a city built around politics, money, power and the grand public institutions designed to reflect America's glory back at itself, the arts establishment is watching to see how the Hirshhorn is going to pull itself out if this mess.

The first step it has taken to that end is Chiu's appointment.

Chiu was born in Darwin in 1972 to Chinese-Australian parents. Her father was a dentist, her mother a dental nurse. She remembers a typical Darwin childhood – a mango tree in the backyard, a multicultural school. She even has foggy memories of surviving Cyclone Tracy, or at least of being evacuated south while the city was made habitable in the storm's wake.

While Darwin might have been one of Australia's most multicultural cities, multiculturalism was not yet part of Australia's language. "My father had an interest in us being Australian," Chiu says. "He is Chinese and when he and my mother were married it was unusual to be bicultural. He wanted us to speak English first. We picked up our Chinese cultural understanding along the way."

Chiu was in her early teens when the family moved to Hunters Hill and she was sent to Loreto Kirribilli, the private Catholic girls school spread out underneath the Harbour Bridge.

On graduation Chiu chose to study art history at the University of Western Sydney rather than follow her peers across the bridge to the University of Sydney. Her parents backed her choice. It was to prove a crucial decision, one that she puts down to "good luck and timing".

The university had only achieved university status a few years earlier in 1989, the same year the Chinese government cracked down on protesters at Tiananmen Square, killing anywhere from 300 to 1000 students in the process.

Then Australian prime minister Bob Hawke wept during a ceremony to mark their deaths and declared that to "crush the spirit and body of youth is to crush the very future of China itself". He extended the visas of about 10,000 students, some of whom ended up in class with Chiu, whose growing interest in contemporary Chinese art was cemented.

Over the coming years, as Paul Keating followed on from Hawke, Chiu benefited from his determination to help – even coerce – Australians into accepting that they were part of Asia. "I was supported by the Australia Council to go do research in China," she says. "I was sent to Japan. I was able to do residencies in tons of places."

Even so, when she voiced her intention to specialise in modern Chinese art, a lecturer told her she was crazy. "At the time the lecturer was actually being responsible. The lecturer was saying. 'Look, is this really a field? How many books have been written on the subject'?" she says. "At the time there were only two."  She was warned she would never get a job.

But Chiu held firm and on graduation was one of the founders of the 4A Centre for Contemporary Asian Art, which is now based in Haymarket.

The gallery and arts centre was launched with the financial assistance of James Fairfax and has proved a remarkable success. Sydney is not always a hospitable environment for galleries and arts centres, and it is certainly not prone to philanthropic ventures, but 4A has become a fixture.

It was during her time with 4A that Chiu first crossed paths with a director from the Asia Society, a New York-based non-profit organisation with satellites around Asia dedicated to building ties between Asia and the US, especially in politics, business, arts and education.

Shortly afterwards in 2001 the Asia Society poached her from 4A and installed her at its imposing headquarters on Manhattan's upper east side overlooking Central Park as its first curator of Asian and Asian-American Art. As The New York Times noted in a profile of Chiu published in 2008, at the time it was the only position of its kind in the US.

In 2004 after curating her first few exhibitions she was promoted to museum director. By then, contemporary Chinese art was becoming the hottest thing in the art world – luck and timing, or foresight – and Chiu was becoming one of its foremost experts in the United States. Her exhibitions were having increasing impact and her reputation in the field across the United States was growing.

She was also demonstrating what impact cultural diplomacy could have when politics failed.

In 2008 the Chinese government quietly decreed that the Cultural Revolution would not be mentioned during its Olympic celebrations. Chiu's exhibition, Art and China's Revolution, which included many artefacts and posters of the period, attracted more visitors than any other in the society's history. In 2011, relations between Pakistan and the US plumbed new lows with what has become known as the "Raymond Davis incident" in which an American "administrative and technical official" attached to the Lahore consulate killed two men. Throughout the crisis, Chiu and Pakistan's ambassador to the United Nation's spoke daily to ensure that an exhibition Hanging Fire, Contemporary Art from Pakistan went ahead.

Chiu's role became increasing national, and international too, as the Asia Society expanded in the US and across the region and as Chiu became an in-demand guest lecturer.

It is said that every American visits Washington, DC, twice, once as child, once with their children. They come to see the Capitol Building, the White House, the monuments to Jefferson, Washington and Lincoln, and to see the museums of the Smithsonian Institution – sometimes referred to as "the nation's attic". It is difficult to exaggerate the reverence and affection Americans have for the Smithsonian.

The Hirshhorn sits between the Air and Space Museum and the Native American Museum, opposite the National Gallery of Art and the National Archives, home to the US Constitution and the Declaration of Independence.

The wheels began to come off at the Hirshhorn when the museum's director, Richard Koshalek, had a really big idea in 2009. The Hirshhorn's purpose-built home is a three-story-high elegant donut. Koshalek planned to install a bubble that would be inflated from the hole in the centre for two months a year to house public events.

The design won awards and the local establishment embraced the idea.

By last year, though, the project's budget had blown out from $5 million to $15 million. Some directors feared focus on the bubble had distracted management from the Hirshhorn's core mission.

Worse, major donations towards its construction had ceased. In May last year the board was split on a vote on whether or not the project should be abandoned and Koshalek resigned. Two members of the board followed him out, bringing the total of resignations from the board over the previous year to seven. Then, the chairwoman of the board, Constance Caplan, quit and her scathing resignation letter found its way into the pages of The Washington Post.

"I now see the Hirshhorn taking, with both overt and tacit approval by the Smithsonian: a regression to programming that imitates a predictable pattern at many other modern and contemporary museums," she wrote. "I see the Hirshhorn abruptly regressing from the vision of serving as 'the nation's museum of contemporary art' – a vision especially appropriate to its splendid, unique setting – retreating at a time when precisely because of the challenges at hand, this larger role is more important than ever."

Earlier this year, when it became clear that he would not be promoted to director, the Hirshhorn's chief curator, Kerry Brougher, quit too, taking a job heading the new Academy of Motion Picture Arts and Sciences Museum.

After his departure the Washington City Paper noted that, "The Hirshhorn will announce its next director in late April or early May. That individual will have a great deal of influence in shaping the future of the Hirshhorn, if a director can be found who will stick to the position."

The announcement that Chiu will take the roll was greeted positively in Washington. The Washington Post's chief art and architecture critic, Philip Kennicott, says she is well known and well regarded by reputation.

He says people want Chiu – and the Hirshhorn – to succeed. They want to see the institution regain its vigour and not just attract significant exhibitions to DC, but to curate and tour its own. Chiu has proved she can do that.

The Wall Street Journal noted approvingly of her appointment in June that she is "known for being a savvy fund-raiser". In its announcement of her appointment the Hirshhorn noted that she had raised 80 per cent of the Asia Society's $29 million budget from private donors and foundations.

"Melissa really fits the bill here," the Smithsonian Undersecretary Richard Kurin, who chaired the search committee, told the Journal. "Melissa comes from Australia, she's of Asian descent and has dealt with the arts of the whole Asian continent, stretching halfway around the world and through the Pacific. The idea is she could help the Hirshhorn really have a global impact."

If Chiu is nervous about the job, she is not showing it. Over a bowl of chicken soup in an elegant New York restaurant, she says in her precise manner that she is looking forward to rebuilding the team.

She expertly dodges questions about the how the organisation has been managed in the recent past, and even finds something positive to say about all the media attention it attracted.

"I actually took that as a positive, people really cared about what happened there."

She talks up the opportunities of being involved in the creation of a new leadership team, just as though it was all good luck and timing.