What version of Windows are you currently running? Win + R and type in "winver" to check. Which update is it attempting to install?
I would recommend running chkdsk /f. Once that is done, you can check the log to: 1) make sure it actually ran, and 2) see if it actually made corrections to the file system. The log is located in the event viewer under the Windows Application log, source Wininit.
Then run the following command: dism /online /cleanup-image /restorehealth to make sure the component store corruption, if any, gets fixed. Then run sfc /scannow.
It makes no sense to me to start troubleshooting Windows Update issues without making sure the OS itself is working properly first. Of course, this is my mentality because I come across about a dozen broken operating systems a day.
If the update troubleshooter doesn't work, I'd try using Complete Internet Repair to resolve the issue. Also, if you have no reason to be using Bitlocker, I'd recommend disabling it since people tend to lose their keys in my experience.