Nalis was the first to react, blasting a right foot shot into the top corner from 12-yards.
Leeds contributed to their downfall with 13 minutes left when Kelly, under no pressure, headed the ball towards his own keeper but only succeeded in putting it past the stranded Neil Sullivan.
Leicester keeper Stuart Taylor, on loan from Arsenal, was well protected as Leeds struggled to break down The Foxes' well organised defence, but he needed a save out of the top drawer to stop Brian Deane's goal-bound header in the 71st minute.
Deane also blasted over the bar from jut five yards a minute later as Leeds lacked the cutting edge.
Taylor moved smartly to palm away David Healy's fifth-minute header for a corner, but he spent much of the first half as a spectator.
Leicester looked more comfortable on the ball and have now lost just once in 11 games.
Healy did have the ball in Leicester's net in the 19th minute but the effort was rightly disallowed as the Northern Ireland international was clearly offside after Sean Gregan and the lively John Oster put him through.
Sullivan made two excellent saves in rapid succession midway through the first half, stretching to tip Joey Gudjonsson's free-kick over the bar and then clawing away David Connolly's right-foot shot after Nikos Dabizas had headed down Gillespie's in-swinging corner.
Oster, whose loan spell from Sunderland has been extended by a month, almost conjured up a goal for Healy in injury time at the end of the first half, skipping between two defenders and crossing to the far post where the forward could not turn quickly enough and was crowded out.
But the Leeds squad clearly needs strengthening and that will happen if, as expected a Yorkshire-based consortium injects cash into the club this week.