I'm not an auto-tech, electrician, etc. - If these instructions don't make sense to you don't use them because they may be nonsense. Use your own best judgment before trying to follow my line of reasoning:
Can you plug a 12VDC source into the trailer's umbilical cord? Have the 120 VAC disconnected and the umbilical disconnected from your tow vehicle before going any further. I'm assuming that you have a standard 7pin connector setup. Pictures of the connector pinouts are available all over Google.
First check to see that your battery isn't dead. Read the voltage between pins 1 and 4 to see what shape your battery is in.
Next check for continuity between pin 3 and pin 1 on the trailer's connector. I would also try placing the meter's ground probe on a clean piece of metal at the tongue of the trailer. If you read anything near 0 Ohms you've probably got a short that you'll have to find. If the number is not near zero you may proceed.
Just connect the hot wire (12VDC) to pin 3 of the trailer connector. Connect the ground to a good trailer ground or pin 1 of the connector. See if you get any lights to illuminate. Remember that some of the bulbs may be bad so it is important to identify which lights if any go on and which ones go out. Let us know what happens.