Neither customer care or technical support are able to resolve my issue

I don't have a valid Windows License on my machine anymore, because Dell made a mistake when replacing my mainboard. Now my machine is out of warranty, and the problem only became apparent when I tried reinstalling Windows with the Dell OS recovery tool.

I called for hours with technical support, they say they're not able to help me or give me a new license. They say they can't escalate the issue, and don't allow me to speak to a manager.
Customer care can't do anything for me.
Technical support tells me to file a complaint, by writing a letter. It took me a while to find the right address, so I mailed a letter three weeks ago. No answer so far.

Nobody seems to care, or able to tell me how to resolve the issue. 
Microsoft support says my Windows product key is blacklisted, and to talk to the people who provided it. Dell says it's Microsoft who needs to resolve my issue.