Carlton Cole and Herita Ilunga both missed glorious opportunities for West Ham in the opening half.
West Ham had a chance to equalise when Kalou fouled Hireta Ilunga but Cech superbly saved Mark Noble's penalty.
But the home side levelled when Chris Burke robbed Ilunga of possession after Ikeme's loose pass and found King.
At the other end Hull allowed Ilunga a free header from a left-wing corner only for the defender to glance badly wide.
Nevertheless, they were able to threaten on occasion, as Ilunga began to find space - but too often found himself lacking in support.
It seemed Chelsea had three points wrapped up but they had to survive a scare when Kalou conceded a penalty for tugging back Ilunga.
But 12 minutes later the Blues were level when King took advantage of a mix-up between Carl Ikeme and Herita Ilunga to score from 12 yards out.
However, luck deserted the Hammers when the England keeper was forced to parry a header from his own team-mate Ilunga and the loose ball was bundled in by McGoldrick.
Bridge will add to manager Avram Grant's defensive options after West Ham's only specialist left-back Herita Ilunga suffered a leg injury in the 1-1 draw with Everton on 28 December.
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Burly journeyman Dele Adebola proved to be a nuisance for defenders Danny Gabbidon and Herita Ilunga, while man-of-the-moment Lewis McGugan and Chris Cohen were beginning to combine well for the visitors on the right.
Having taken an early lead, Stoke seemed happy to soak up the West Ham punches and they were lucky to get away with it when Herita Ilunga found a way through on the left wing and laid the ball back into a crowded box, somehow missing all his team-mates.
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