Considering the state we're in, I vote for a Vatican-style conclave
It works in the Vatican, perhaps it's time to swing it into operation at Dáil Éireann. To elect a pope, cardinals are sealed into the Sistine Chapel where voting takes place twice a day, every morning and evening.
And there they stay, under lock and key, until they reach a conclusion. Concentrates the mind, I expect.
So, with thanks to the Holy See for devising such an efficient system, I'm inclined towards a similar plan: corral all 158 TDs into a corner of Leinster House and impose twice-daily votes for Taoiseach on them. No exit allowed until they send up white smoke signals.