This was a painstaking experience because it took me countless forum posts that mostly never worked for my scenario until I came across one that did the trick. And the post topic was not even about having problems signing in to a Microsoft account from a local account in Windows 10.

What did solve for me was by executing a command in Powershell for a broken Start Menu (since this is also where Cortana can be found at first run).

1. Open Windows PowerShell as Administrator
   1. Right click Start button
   2. Choose Command prompt (Admin)
   3. Key in Powershell in the black window and hit Enter
2. Paste the following command in the Administrator: Windows PowerShell window and press Enter key:
   
   Get-AppXPackage -AllUsers | Foreach {Add-AppxPackage -DisableDevelopmentMode -Register "$($_.InstallLocation)\AppXManifest.xml"}

   1	Get-AppXPackage -AllUsers | Foreach {Add-AppxPackage -DisableDevelopmentMode -Register "$($_.InstallLocation)\AppXManifest.xml"}

Do not worry. While the command gets executed, your Start Menu won’t actually be broken. You can still use it as you want but if you wish to wait it out, no problem. It won’t take that long anyway.

After the command has finished, I was then able to log in to my Microsoft Account.