Yakubu's second goal since his arrival on loan from Premier League side Everton and King's fourth goal in his last three league games gave the Foxes a vital three points as they moved up to seventh place in the Championship.
Sven-Goran Eriksson's side's push for the play-offs continues to gather pace after their fourth successive win leaves them unbeaten in seven league games.
The home defeat for Derby piles more pressure on boss Nigel Clough and leaves the Rams languishing in 17th place in the table, without a win in six league games.
Leicester looked the more livelier of the two teams in the opening exchanges and after Paul Gallagher sent Patrick Van Aanholt through with a clever pass that split the Rams defence, the Chelsea loanee fizzed the ball across the face of the goal but Yakubu could not stretch far enough to convert from close range.
Darius Vassell thought he had opened the scoring on 21 minutes when his 18-yard shot nestled into the top corner, only for the referee to disallow the effort for offside.
The visitors did break the deadlock six minutes later when Yakubu took advantage of some poor Derby defending to fire into the top corner with a smart left-footed volley on the turn.
Gallagher almost doubled the Foxes' advantage on the half hour but his curling shot from the edge of the box flew agonisingly past the post.
The Championship player of the month, Richie Wellens, had a shot blocked as Leicester started to dominate an under-confident Rams outfit.
The home side showed no signs of troubling debutant Pereira Ricardo in the Leicester goal as the visitors did all the pressing in the first period and were rewarded with their second goal on the stroke of half-time.
King latched on to a perfectly weighted through ball from Gallagher to slot past Stephen Bywater following impressive build-up play involving Van Aanholt and Yakubu.
Derby were disappointed not to be awarded a penalty a minute after the restart when Steve Davies' free-kick was blocked by a Leicester player's hand in the area but the referee waved play on.
The game was held up for five minutes as central defenders Ben Mee and Souleymane Bamba both received treatment after clashing heads on the edge of their penalty area but when the contest restarted, the Foxes showed no signs of stopping as they passed the ball around fluently.
The Rams, who lacked creativity in the final third, tried in vain to find a way back into the game but were thwarted by an impressive Foxes backline, who were influential in Leicester's victory.