After Kun was removed on October 25, two attempts were made to elect a replacement from two candidates - opposition MP David Adeang and the government's Rennier Gadabu.
On each occasion, the vote was nine-nine so Speaker Marcus Stephen, who does not get a vote in presidential elections, called for a third attempt to be held on Monday, October 30.
The new president will be the island's fourth leader in four years.
Adeang was a long-time finance minister in Nauru and along with former president, Baron Waqa, was accused of receiving kickbacks from a phosphate exporting company.
They have both denied they were involved and although an Australian Federal Police investigation into the event was launched in 2015 it is unclear what the outcome of that investigation was.
Both men were also central to the suspension of the opposition from parliament and the subsequent manipulation of the judicial system which an Australian law professor called a shameful episode.
Professor George Newhouse called on the Pacific to respond to the damage being done to Nauru's democracy and its judiciary, but it largely stayed mum.
Baron Waqa is next month due to assume the secretary general's position at the Pacific Island's Forum, though that appointment has since come under scrutiny from some leaders.