WARNING: Several people have bricked their device by following this particular guide for devices running 18.104.22.168 thru 22.214.171.124. This is the one and only rooting guide on this site that I have not personally tested. Because of this, you should seriously consider letting your device update to version 5.0.5, as the next paragraph suggests, and root using the method for that software version.
The Fire TV 1 running software versions 126.96.36.199, 188.8.131.52, 184.108.40.206, 220.127.116.11, 18.104.22.168, 22.214.171.124, 126.96.36.199, 188.8.131.52, and 184.108.40.206 is rootable, but because your eFuse is blown with those software versions, you will not be able to unlock your bootloader. An unlocked bootloader helps you restore your device if you brick it in some instances, but it’s not necessary to root. Since you can’t unlock your bootloader anyway, it’s best to update to version 5.0.5 (Build 537174420) if you can to make the rooting process easier. Let your Fire TV start downloading an update and if it begins downloading version 5.0.5 (Build 537174420), allow it to update and then follow this guide. If it begins downloading any other software version, then pull the power to stop the update from downloading, and follow these steps to root: