The February 26 election resulted in political stalemate and there is little sign of the logjam being broken, according to lawmakers from all sides.
Neither Fine Gael or Fianna Fail, the two main parties, have the numbers to form a government without the support -- active or passive -- of the other.
The election of a prime minister, or taoiseach, is a prerequisite to agreeing the composition of the next government.
Kenny's Fine Gael party won 50 seats in the vote -- 27 fewer than in the previous election in 2011.
Its bitter centre-right rival, Fianna Fail, increased its vote substantially to 44 in the 158-seat Dail, the lower house of parliament.
In today's second vote for prime minister, 51 lawmakers voted for Kenny, fewer than last time; 43 voted for Fianna Fail leader Micheal Martin and 10 for Ruth Coppinger, a deputy representing the left-wing Anti-Austerity alliance who became the first woman nominee for taoiseach.
The two main parties appear ideologically similar but there is a deep mistrust emanating from their evolution as opponents during the country's bitter civil war in 1922-23.
Since then, one or other of them has held power, usually as the main partner in a coalition arrangement but never as part of the same government.
The pair must now decide to form an historic pact or be prepared to trigger another election and risk public anger over their failure to do a deal, according to political analysts.
The pressure has been mounting for the two biggest parties to form a government in recent weeks amid ongoing crises in healthcare and housing.
Following the second taoiseach vote, Kenny and Martin were expected to begin face-to-face exploratory talks on government formation later today.
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