Am I out of luck with my motherboard change and no Win8 key?

My motherboard stopped working in my Dell Inspiron 660S desktop.  It was dead.

Before it died I had done the free Win10 upgrade from the original Win8 Dell OS.

I bought an identical board (brand and model), swapped the old Intel chip into it and had to use the rescue disc to boot to Windows.  So it reset Windows from the rescue disc.  

Now I have no access to my Win8 key since it is on the dead board.

I talked to Microsoft and they tried but said I am out of luck, the Win8 key is needed to activate the Win10 upgrade, and the key is on the dead board.  I have to buy a new Win10 copy.

Am I out of luck here?  The computer is out of warranty (2013).

Thanks.

Alan