More than anything, this was a bit of a story conceit on Miller's part; he was a very strong Batman fan, and took some personal issue with Superman's near indestructibility. This was one reason why he allowed the fight between the two friends to take on such a brutal tone in his graphic novel. Visually, it offered a lot of pathos, as well; a still-healing Clark coming to pay his respects to "the only man that ever truly beat you" [Bruce's words to him in the comic] was both beautiful, and evocative.
In story, however, the answer provided by Fuzzyboots was correct; between the dirty missile blast, the lack of solar radiation and the powdered kryptonite traces still in his body, his healing may not be functioning at optimum. Keep in mind, this would like be the early version of Jon Byrne's Post-Crisis Superman, who was totally solar powered, and shown to be markedly less powerful------ and consequently, more vulnerable---- than previous versions. Though to be fair, up to that point  we'd never seen even that Superman have a bruise for more than a few hours.
A better question would be why, after being affected by the nuke, didn't Superman just fly out into space and directly absorb some sunlight and come back fully healed and recharged, or at least go under some special solar lamps for the boost, especially before he had to face Batman in that no holds barred fight?