Try to run the Windows update troubleshooter and see if it fix the issue:
* Press Windows Key then type ‘troubleshoot’ and hit Enter
* Select “View all” on the left panel and run “Windows update” troubleshooter from the list
* Click Advanced, ensure to check Apply repairs automatically and also select Run as administrator
* Click Next and follow the on-screen instructions to complete the troubleshooting.
Once done, check if it allows you install Windows Updates without any errors.
a. Press Windows Key + X.
b. Click on ‘Command Prompt (Admin)
Note: Click ‘Yes’ if it prompts for UAC(User Account Control).
c. Type following commands into the command prompt one after the other
net stop wuauserv
net stop bits
net stop cryptsvc
Keep Command Prompt window opened.
Rename Windows Update folders
At the command prompt, type the following commands, and then press Enter after each command:
ren %systemroot%\System32\Catroot2 Catroot2.old
ren %systemroot%\SoftwareDistribution SoftwareDistribution.old
Then type the following, commands and then press ENTER after each command:
net start wuauserv
net start bits
net start cryptsvc
Close the Command Prompt window and check if you can install the updates.
At the risk of sounding a little ridiculous, make sure that your time and date are correct on your computer. An incorrect time, even if it's off by a few minutes, can wreak havoc on updates.
If you aren't able to update it with the direct link update provided by CORROSIVE, you may end up just having to do a clean install of 10. If you updated to Windows 10 during the free update offer, your system has a Windows 10 digital entitlement, so all you would have to do is a clean install of Windows 10 and it would activate due to that entitlement. If you have to do that, make sure you save all of the stuff that you need to so you don't have to download it all again afterward, plus save the things you would lose completely.
Edit - I hadn't refreshed for a little bit, so I didn't see Amanda's subsequent additions to the thread after CORROSIVE's. Check them out.