Second-half goals from Shinji Okazaki and Islam Slimani heaped more pressure on the German manager on Tuesday night.
The Reds have now conceded 10 goals in their last four matches and Klopp was enraged with their lack of defensive solidarity.
“It’s easy to explain,” said Klopp. “We played really good football in the first half and that should have decided the game.
“We had big chances, we had strange situations.
“But football is a 90-minute game and it’s the second ball after a corner I think and a throw in again.
Liverpool dominated the first half, only for Klopp to replace Philippe Coutinho with Ben Woodburn at the break.
But the young Welsh talent wasn’t able to exert the same influence in midfield and Liverpool’s defence were unable to keep Leicester at bay.
“I’m sick of goals like this to be honest,” Klopp continued.
“You cannot play football and, in this case this Leicester team somehow survive in the game because we are not clinical enough.
“Sometimes that’s OK but we should have concentrated until the end. They scored too easily.
“The first goal we didn’t push up together. I can say it one time or one thousand times.
“It’s not nice conceding these goals today again because of this.”
Liverpool return to the King Power Stadium on Saturday to play Leicester in the Premier League.
Meanwhile, Klopp’s odds to be the next Premier League manager sacked have been slashed from 66/1 to 33/1 in the wake of the 2-0 defeat.
The former Borussia Dortmund boss now has a win percentage of just 50.9, lower than David Moyes during his time at Manchester United.