How do the prisoners in Night, including rabbis, reconcile their agony with their faith?

Depending on which rabbi or religious mentor one studies from Night, the answer to your question will vary, because some of the Jewish leaders were able to reconcile their faith with the mind-numbing experiences of the concentration camps, but others were not and lost their sanity.

For example, Moshe the Beadle, while not a rabbi, had such strong faith at the book's beginning that he mentored Elie in the Cabbalist form of Judaism--a sect that arguably requires more faith than more orthodox forms of Judaism because of its mystical nature and emphasis on man's ability to achieve perfection.  But, whenMoshe escaped from deportation/death, he was a hollow man whose goal was simply to warn others that all Jews weregoing to experience incomprehensible horro. He no longer had the faith to mentor Elie or even encourage him.

Another religious leader, Akiba Drumer, whom Elie meets in the camps, serves as a kind of replacement for Moche as a mentor for Elie.  He keeps Jewish traditions and encourages other prisoners by telling them that God has not forsaken them and that either their experience is a test or a time for them to repent of their wayward ways and return to their faith. Later, though, when Drumer's health is failing and he realizes that he is going to be sent to the death camp, he simply asks for Elie and others to sing the Jewish death prayer for him.  Elie describes his eyes as "open wounds."  So while he might have kept his faith, the agony of the camps certainly takes away the hope associated with his beliefs.

A third description is Wiesel's description of Rabbi Elihaou on the torturous run from camp to camp in the dead of winter.  The Rabbi believes so strongly in survival and his son's loyalty that he does not realize that his own son has abandoned him.  Elie does not have the heart to tell him that his son saw him lying in the road and ran on without him. Ironically, the rabbi does not lose his faith; he simply keeps believing that his son and God would never leave him.

Finally, Elie himself had wanted to be a rabbi, but the agony of his Holocaust experience robbed him of his faith.  He becomes angry with God and describes his faith as lost when he witnesses babies being thrown into the ovens. At the book's end, Elie describes himself as having "corpse eyes" when he looks into the mirror.