Plug the other end of the cable into another computer. The system controller's memory appears as a virtual flash drive on that computer.

Open a terminal and navigate to the reform2-lpc-fw folder you've downloaded from the Reform repository.

Check lsblk or dmesg -w to find out which drive the LPC shows up as, then edit flash.sh with the correct path to the virtual flash drive (e.g. /dev/sdx).

Execute flash.sh as root.