There are lots of great LED lighting guides here on Instructables so I won't waste time on what they cover. I will reference a few during the course of this instructable that I thought were good like The definitive guide to LED accent lighting.
There were a few options online for safe lights and some were even LED ones but most didn't offer what I wanted. I wanted red lights for at night in case I needed to access the safe without totally ruining my night vision and white lights for other times when I wanted to be able to see in the safe easily. I wanted something that the bulbs wouldn't need replaced too often and that was flexible enough to light up the inside of the safe easily. I chose the SMD 5050 300 strand in white and the SMD 5050 150 strand in Red. I also wanted there to be a motion sensor in the mix so I wouldn't have to try and line up a door switch or worry if I shut off the lights all the time.
(1) Warm white 5M SMD 5050 LED Strips Light 300 Led flexible Car Auto 60 leds /Meter - $13
(1) Red 5M 5050 SMD 150 LED Strips light leds 30 LEDs /Meter Flexible - $11
(1) Micro Switch Basic Snap Action Switch 15A V-15-1C25 - .50 cents
(1) 4x2x1 project enclosure. - $4
(1) 2.1mm x 5.5mm Female CCTV Power Jack Adapter - .54 cents
(2) 3/4 in. x 3/4 in. x 8 ft. MDF Primed Inside Corner Moulding - $6.60
(1) DC 12V 3A 36W US Power Supply Adapter for 3528 5050 RGB SMD LED Lamp Light Strip - $12
(1) Westek MLC12BC-4 Indoor Plug-In Corded Motion Activated Light Control - $18
Total Cost ~ $68. I'm sure you could get some of the items cheaper.
60/40 rosin core solder
Hot Glue Gun and glue sticks
Step 1: Mounting the Light Strips
I was looking for a way to mount the strips of LEDs so they would be at an angle facing into the safe. I could have run the light strips flat against the inside of the safe door frame but felt directing the lights into the center of the safe would give me better lighting. I took the two strips of molding, cut off a small piece and mounted that to the router guide with screws. This piece would allow me to push the rest of the molding across the cutter at the same angle. All the flying plastic shavings made a huge mess. **Wear Goggles.**
Basically I was taking the edges off the L shape and making it more flattened. I cut the Molding into (2) 54 inch pieces and a 24 inch piece to fit the safe sides and bottom. Unfortunately the pics of the finished molding didn't turn out.
As an alternative to the molding you could tape a piece of flat plastic or whatever across the corners on the inside door frame of the safe to mount the lights too. As long as you are angling the light into more of the center of the safe.
Step 2: Adding the Light Strips
The light strips come in 5 meter lengths. I cut the light strips to match the(2) 54 inch pieces and (1) 24 inch piece. There are segments every 3 lights where you can cut the strips. The light strips come with adhesive on the back so it was just a matter of peeling off the backing and attaching them to the molding. In the picture you can see I ran the red and white lights next to each other the whole way. Also is a picture of a connector I bought thinking it would save me time attaching the strips but I must have bought the wrong ones because they didn't line up with the positive and ground connectors. I ended up soldering the strips together. The strips of lights are marked at every connection with + for positive and G for ground. Just be careful as you go to keep it all lined up. I ran my both strips positive in the center and grounds on the outside corners of the molding. There is a good instructable here for cutting the strips and soldering them. Interactive LED Lab coat
After I had the strips of lights attached to the molding and the connections soldered. I put the whole assembly into the safe facing in from the door frame. And added a few drops of hot glue to the molding to keep it from moving in the safe.
Step 3: Adding the Project Box
Next I wanted to have the power connect to the switch to control the red or white lights. I used a small project box and drilled two holes. For the switch I used a 1/4 inch bit and drilled a whole in the bottom of the project box. I used a 3/8 inch bit to make a whole to put the female power connector through. I then cut a notch in the side of the box to run the wires from the switch to the lights. The switch has a threaded post that you can attach with a nut to the project box. The female power adapter I used hot glue to keep in place through the hole in the box.
I connected the power from the Female connector to the center poles of the switch, then connected the two light strips to the end poles. Inserted all of it in the case. Hot glued the case to the inside of the safe. Made a quick test to be sure it worked and screwed on the cover to the case.
Step 4: Adding the Power and Wire Management.
Once the project box was installed with the switch, wires and female connector inside it. I hot glued the power supply to the inside top right of the safe to keep it up and out of the way. I glued it so the green power indicator light faces out just in case I need to troubleshoot in the future.
I then rolled up and wire tied all the cables for the Power supply and motion detector. I plugged the power supply into the motion detector plug then plugged that into the power outlet in the back of the safe. It slightly covers the USB and network connector in the safe but not enough to make them unusable.