Remove
the led with a soldering iron and attach a pair of wires. I used
a Molex connector , but a pair of wire nuts could be used.
For more about what's inside a Garden Light view my PDF "Garden Light Autopsy" .

Attach a twisted pair of wires to the connector and run down to your
house.
The
wire
runs in the PVC pipe holder, then down the vent of the cooler.

The led is soldered to a short pair of wires to another connector.
Shrink tubing is
used to cover the solder joint (unshrunked in photo).
The led connects to the end of the wire in the house.
Molex 03-06-2023 plug (shown above), 02-0602103 pins. Mates with
Molex 03-06-1023 recpt, 02-06-1103 sockets.
You can get these at Digikey.
Or wire
nuts can be used instead of connectors.
I have the led and wire brought through the cealing vent of the
cooler and use it to light up my Hallway at night.

The lights come on at night and illuminate the hallway with a soft
yellow light..
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