Hi Luc
This is the process you should use . . .
Create bootable Windows 10 installation media on a flash drive on the other PC (see NOTE below)
Pop the external drive out of its casing and put that into the laptop that is not working
Then install Windows on that laptop using the installation media you created . . .
NOTE
You can reinstall Windows 10 at any time and it will not cost you anything !
On the other PC, click this link:
https://www.microsoft.com/en-us/software-downlo...
to download the Media Creation Tool (Click on Download Tool Now), with that you can download the latest Windows 10 ISO (Select Create Installation Media for Another PC), you can create a bootable USB flash drive (min 8GB) using that tool
Insert the SSD into the laptop that is not working
Then, Boot your the from the Installation Media you just created (change Boot Order in your BIOS) to begin installing Windows 10
Since you previously had Windows 10 installed and activated on that PC during the installation process skip the steps which ask for a product key and select the option 'I am Reinstalling Windows 10 on this PC', and activation will not be an issue, your PC will have a digital entitlement stored on the Microsoft Activation Servers
Make sure you install the same Edition of Windows 10 (Home, Pro . . . etc.) you previously had installed to avoid Activation issues