This caught my interest as I have a similar upcoming project idea brewing for my JKU.
I believe your problem lies with the CAN bus that monitors and controls all things in cars these days. It's probably detecting an over-current condition (more current in the circuit than it expects to see) and shutting it down. That's the CAN bus monitor's job. But all is not lost...
Your LED light strips are probably pulling around 400 or 500 milliamps (~1/2A) per meter, from a similar-sounding strip I picked up. If you have a couple meters worth, you've added (guessing) 2 to 4A. Not a ton, but probably well over the CAN bus "Warning, Will Robinson" threshold. Please look up the current draw per meter of -your- new lighting. Your fuse will need to be around 2x the total (ballpark).
You probably need to bring a new circuit wire (I like 16 gauge wire for this) from you battery, through a new 5 or 10A waterproof mini-blade-style fuse holder, poke through the firewall up behind the glove box, into a new 12V automotive relay. Connect to pin 30 and go out [pin 87 "Normally Open"] to feed the LED circuit power. Use the power that you have going to the LEDs today to drive the switch inside the relay [pins 85 and 86, pick either one, and connect the other one to ground]. Make sure the new wires are secured (zip ties) so they don't dangle and electrical-tape the connections so they don't short out against anything. You can get wire and a fuse holder and relay at most auto parts stores, maybe even WallyMart.
The idea is to get the new lighting's power draw off the circuit that feeds the factory lighting. The switch inside the relay draws negligible current, so the CAN bus won't even know it's there.
There are some great articles about relay use floating around. I thought this one was quite to-the-point: Understanding Automotive Relays | Installing Engine Relay
I do hope that works, and that the task not too daunting. If a dumb software guy like me can follow it, I suspect you can too.
All the best.