Hi Freya,
Welcome to Microsoft Community.
I understand that you have an issue with installing Windows Updates which gives an error 80070008. We apologize for the inconvenience caused to you.
I would like to know the detail below to assist you better.
- Did you make any changes to the computer prior to the issue?
Hence you have performed most of the troubleshooting steps. I suggest you to refer the section "Manually reset Windows Update components" from the below article, perform the steps in it and check if it helps. Resetting Windows Update Components will fix corrupt Windows Update Components and help you to install the Windows Updates.
How do I reset Windows Update components?
https://support.microsoft.com/kb/971058/en-us
Registry disclaimer
To do so: Important this section, method, or task contains steps that tell you how to modify the registry. However, serious problems might occur if you modify the registry incorrectly. Therefore, make sure that you follow these steps carefully. For added protection, back up the registry before you modify it. Then, you can restore the registry if a problem occurs. For more information about how to back up and restore the registry, click the following article number to view the article in the Microsoft Knowledge Base: 322756 (*http://support.microsoft.com/kb/322756/*)
How to back up and restore the registry in Windows
Now try installing the Windows Updates and check if can complete it.
Keep us posted on the issue to assist you better.