check your fuses, there should probably be a DOME fuse, I would start under the hood in the fuse box, but could also probably be in drivers side dash panel fuse box as well.
Do you have a multi-meter? I see 3 wires in your picture, which makes sense, because the dome light has 3 position: off, always on, on with door open. That means that one of those wires is a ground wire, the other 2 are your power wires. You just have to determine which is the Always on power and which is the Door Open power wire. You could tap into either one depending on how your want your lift gate light to work, or you could tap into both and have the same function as the factory dome light, it will mimic the rear dome switch position.
Hard to tell from the picture, but you could probably figure out which is the grounding wire just by looking at the markings. There's a chance that both power wires are marked the same and you have 1 different one, which woud likely be the ground. If they aren't, bust out the multimeter and see which 2 wires give you a voltage in the ON position, and which two give you a voltage in the DOOR position. A crude way of checking this is directly with your LED light but I would suggest metering it out