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.
Source: PA
Source: PA