When you upgraded your PC to windows 10, Microsoft recorded the licence key + the machine key (a code that matches parts in old PC) + the email address you gave them to register your upgrade, on a server so if you install Win 10 on same PC again it will always activate. So if all you changed was the hdd, it would auto activate... but you say the motherboard died as well?
In the past, prior to the Anniversary edition of win 10 (that came out 5 months ago so you should have it by now), changing motherboards would deactivate Windows 10 and could mean a new Licence is needed, but since the update you can now move your licence to a new motherboard. Now whether it will happen or not is another question. If the user you use to login is the email address that you gave them to register Win 10, then your licence key is already linked to your email address. Otherwise its still linked to the old PC.
What I would do is download the
Windows 10 media creation tool and use it to make a win 10 installer on USB, as it is latest version of win 10 and supports the move.
When you reach screen of installer asking for licence, click "I don't have a key" and win 10 will continue the process.
Once you have it all installed, and its shows as uniactivated, then follow this
Guide and it will show you how to reactivate PC. If it works then it will move licence to new PC, otherwise you will have to contact MS and explain what you did, they should fix it up for you.