Seen a lot of people online having trouble with this one. Seems for some people Kent dies almost as soon as they enter the hospital even though they go there as soon as they are prompted to do so so its not like the timer runs out. Lots of people just saying "oh its a bug" or "load an earlier save and hope it doesn't happen again" but heres something that has worked for me on a few playthroughs that I cant see anyone else online posting.
For whatever reason Sinjin and his henchmen go hostile and kill Kent, this is caused by them reacting to the robots (the mr handy nurses and protecterons) on the upper floors. You should kill one of the henchmen (either Northy or Smiling Kate), go to the hospital and kill every robot in there before killing the other henchman. Now you can go to the hospital and make it to Kent without Sinjin going hostile and killing him as soon as you enter the building.
Hope this helps.