A last resort might be to perform and in-place upgrade repair. That is to say...... upgrade it again, overtop of itself from within your current install, just double click setup.exe in the installation media. Just be sure to check the little box to get updates as part of the upgrade process.
That will generally keep all your programs, data, drivers and such safe, but.....
I would still create a disk image before going forward.
Installation media can be obtained here (probably a good idea, especially if your ISO / installation media was obtained / generated more than a couple months ago).
http://www.microsoft.com/en-us/software-download/techbench
If you already have the ISO you can simply mount it from within Windows 10 and run setup.exe from there. Again, having the latest ISO from which to produce the installation media will put you ahead of the game here.
Neosmart has a pretty good article on it here https://neosmart.net/wiki/windows-10-repair-installation/
You can skip the first couple paragraphs where they are promoting their recovery CDs and drill down to the meat of the article below.
Deleting the folder you mention was a Windows 7 trick. In Windows 8 and 10 the DISM should be able to repair certain problems with your system.
Administrative command prompt and type:
DISM /online /Cleanup-Image /RestoreHealth
You may get a message about not being able to find a file needed but that usually happens with one of the known problematic files.
If that doesn't help with the updates you might try this.
http://windows.microsoft.com/en-us/windows/troubleshoot-problems-installing-updates#1TC=windows-10
Make sure you restart you system several times a day and even shutdown before restarting, if you can, while trying to get updates. Sometimes one update has to finish installing before the system will continue.
If you are still trying to get updates go back and do a manual check for updates. If you get some type of error message or nothing happens, restart the system.
Then give it an hour or at least thirty minutes and try again. If you don't see any progress in the next 24 hours run the troubleshooter.
What worked, the manual updates or the troubleshooter?
If it was the troubleshooter perhaps it created a log of what was done.
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