You do not need the key code, as it is embedded in the firmware. Just install Windows 8.1 Home with Bing (as opposed to Windows 8.1 Home, which you do not have a key code for) and it will work. If you did not back up Windows 8.1 Home with Bing from your stick then there is no legal download available for that version. So, you will need to install Win10 Home (assuming you activated W8.1 before you wiped it, otherwise you're knackered except for returning to Intel/illicit downloads).
To install Win10 clean, install the pre November build of Win10 (still available as an ISO download from MS, you'll have to search for it), skipping the key code entry every time you're asked, then turn off write caching on your SSD (assuming it is the Kingston SSD, the one that causes the W10 crashes). If you check, it will show as 'Activated' but this may not mean a proper activation due to MS confusing approaches. Then update to the latest version through Windows Update. The latest version will activate using your Win8.1 key from the firmware, and will show up as being activated with a 'Digital Entitlement'. You now have your W10 licence in place.
If you wiped Windows® 8.1 operating system and the recovery partition from the eMMC then, installed Windows® 10, there is no way to get the product key. It is necessary to get in touch with Contact Support and request a replacement. Microsoft® policy does not allow us to send a product key.
The replacement unit will come with Windows® 10 with November update included.