doady: Both my computer and my dad's computer had this problem. You can see because an entire core/thread of CPU is constantly being used in task manager.
I was able to fix this by downloading and installing the KB3172605 update mentioned above by groundhog42. But I had restart the computer and then run it right after restart, otherwise it still wouldn't update properly. Then after WIndows can find and download other updates normally, and CPU usage is now back to normal too.
I had similar problems a few months ago, basically, since I wasn't completely patched up prior to them switching to monthly updates, the updater was having serious issues finding the necessary patches.
For me, switching to WSUS offline up date and updating that seemed to fix the problem, but if there's anything wrong with the database that MS uses to store that information or the disk itself, you might have issues.