How many rads of radiation do you think Kirk got in the warp core? During the Wood River Junction criticality accident, a worker got over 10,000 rads and died in 49 hours. That's the highest dose in the real world that I know of (aside from Hiroshima and Nagasaki). Kirk died only a few minutes after crawling out of the core. He didn't even have time to vomit (the first symptom of radiation poisoning). He didn't have time for his skin to blacken, his hair to fall out, etc. He just passed away. How many rads do you think he got and what do you think was his exact cause of death (too soon for anemia)?
It was magic plot and drama sensitive warp engine radiation. It kills by getting you right in the feels. And that Jeffries tube Kirk crawled through to get to and from the core? Full of puke. That's why he closed the hatch when Spock showed up, to try and cut off the stench.
Probably anywhere between 300 and 500 Grays (30 to 50 kilorads), enough that the onset of acute radiation sickness would manifest almost immediately, followed by loss of consciousness minutes later. That much radiation would disrupt the central nervous system and lead to death before any other body tissues even have time to fail; Kirk's heart would probably stop before nausea had even fully set in.