2016 (even 2012 / 2019 possibly also on newer editions)

Sometimes it can happen that a key entry is corrupt and you can't activate your Windows anymore and even worse trough the GUI you can't change the product key or reinstall it.

So the only option is an elevated Powershell, like so often...

Remove the current key with: 

slmgr -upk

Install the new key (even if identical to the already installed one)

slmgr -ipk yourkeyhere

It may take now 2-3 minutes until your installation is activate