Hart failed to hold onto an innocuous cross from Paul Gallagher - and King converted the rebound midway through the second half.
Sol Bamba gave the Championship side a first minute lead but James Milner and Carlos Tevez put the visitors in the ascendancy before half-time.
Leicester earned their second chance and were a constant threat from the set-pieces of Gallagher. New signing Bamba troubled Roberto Mancini's side in the air and Hart made some fine saves before his blunder.
The game got off to a dramatic start with the home side taking an early lead. Former City striker Darius Vassell won the corner and Gallagher's centre was flicked on at the near post before Bamba applied the finishing touch from close range.
Vassell was causing problems with his pace and earned a free-kick just outside the box when brought down by Adam Johnson.
Lloyd Dyer whipped his cross into the far post and Jack Hobbs powered a header back across goal which Hart was forced to push into the air before collecting at the second attempt.
Roberto Mancini's Premier League side tried to find their composure and Kolo Toure's through-ball almost played in Jo.
But the ball had just too much pace for the striker to seize control and ran through to Foxes keeper Chris Weale.
Johnson's centre picked out Patrick Vieira and his shot from eight yards out was blocked by Weale, before Leicester regained their momentum and caused all sorts of problems from a series of set-pieces.
Hart made a superb save to turn aside a 25-yard drive from Leicester midfielder Yuki Abe destined for the top corner.
From the resulting corner by Gallagher, Hart again showed his worth by parrying a close range header from Bamba, who had another header blocked on the line by Shaun Wright-Phillips.
Back came the visitors and Johnson should have scored from close range but deflected his shot around the post after failing to make a full contact.
Leicester continued to pose problems and Richie Wellens played in skipper Andy King. He tried to chip Hart, who had raced off his line to narrow the angle, and failed to clear the keeper.
Then in the closing seconds of the first half an audacious finish from Tevez put his side in front.
Milner supplied a low centre and the Argentinian striker had the presence of mind to backheel the ball past Weale.
Leicester began the second half on the offensive and Dyer was sent free of the visitors' defence by Gallagher - but he lacked composure and shot wide.
Sven-Goran Eriksson's side looked to have regained their belief and continued to pose problems from set-pieces.
Yet another Gallagher corner picked out a Foxes player and this time Hart fisted away a header from King.
Hobbs should have done better when he headed over close range from Gallagher's centre, before Weale went down to get his body behind Milner's low drive from the left side of the box.
Then a blunder from Hart enabled King to bring Leicester back on level terms after 64 minutes.
Gallagher - once again - curled in a centre from the left which bounced just in front of Hart. However, the England international goalkeeper failed to grasp hold of the ball and the alert King snapped up the rebound.
Leicester were in the ascendancy and Wellens was brought down right on the edge of the box by Nigel De Jong, before Mancini made what appeared to be a defensive minded switch with Jo replaced by Pablo Zabaleta.
Milner came to his side's rescue when he headed Bamba's header off the line - but back came the visitors and Weale made a great save to deny De Jong from close range after he had been teed up by Tevez.