Hi Pétur Harðarson,
Thank you for reaching out to Microsoft Community. We are sorry to hear about what happen to your Surface Headphones. Definitely this is not the experience that we want you to have. We appreciate the time and efforts you used to troubleshoot and try to resolve the issue.
It may seem that this is a hardware issue already. For replacement of the Surface Headphones, we recommend that you contact back the point of sale (Amazon) to have it replaced.
If this answer your question, we highly encourage you to inform our Community by clicking on Yes or marking my post as an answer. This will help other members who may have the same issue as you. Otherwise, don't hesitate to update us so we can further assist you.
Kind Regards,
Marrion Y.