To anyone curious, there is a brown wire which wraps along the edges of the keyboard and attaches to the board using a screw labeled “PROXY” near the power switch.
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Avoid taking this keyboard apart, as the whole backside is not only glued, but tampering with the wire in the slightest confuses the device which will cause the backlight to keep coming on and off constantly with no hands present nearby. I had no choice as I opened it up to clean out a spill, ideally it would need to be recalibrated, but good luck with that.
I love this keyboard, but the lack of an always-on backlight mode despite countless customer complaints and Logitech acknowledging the issue is just disappointing. The constant misfiring of the backlight drove me crazy as well so I just ended up disconnecting the wire, it still lights up but only while typing.