Windows 8.1 update search never ends

Disclaimer

This question has been asked so many times already, but none of the solutions worked, so here we go again.

Introduction

A friend of mine had a problem with her Laptop. So I re-installed Windows 8.1 and wanted to search for updates, but it never found updates. At first I thought it was just taking long too load and said to her she should do this again when she's home.

A few days later she sends me a message that she can't connect her TV via HDMI (the Laptop never sees the other display, not even in the settings) and that also all her functional keys (brightness, etc.) don't work. I asked her to show me dxdiag (because I wanted to see if she can install drivers if it's an Nvidia gpu for example) and saw that she had sth. along "microsoft basic display adapter", which I found out will be displayed if no drivers are installed. So I wanted to do the update search again, however, like before it never finds updates (not even after hours)

Research

So I started researching a bit and see this is a really common problem. I have now tested so many things out, I don't know what else to do. I did

• Run the Windows Update Troubleshooter, which found problems and sait it fixed them. The error code was/is 0x80070490
• Tried to install this (3173424) and this(3172614) driver manually, however then it's always stuck at sth. along the lines (it's in German, so only rough translation) Windows Update standalone installer for updats on this computer and like before, never ends
• Tried to install this(3138615) driver, same result as above
• Do the listed steps found in this answer and this answer
• Do the listed steps found in this article
• Tried to use autopatcher -> No result
• Tried to use portableupdate -> No result

I was also able to copy the contents of her log to my computer (I'm doing all these steps via TeamViewer) which you can find here (care, big file)

The last thing I would try to do is to re-install Windows 8.1 again or just go straight to Windows 10, however we don't live THAT close together and she and I work a lot so it will be hard to find a date. So the preferred solution would be that I can somehow fix it via Teamviewer. I'm nearly giving up, but maybe you have some ideas?

I've been able to fix similar issues using https://www.wsusoffline.net in the past. An outdated windows update agent was unable to correctly communicate with the update servers. Wsusoffline was able to install the required updates fixing the windows update agent communication issues. No guarantees it will help in your case but definetly worth a try.