I limited experience with electric trailer jacks. Plenty of electrical, 12 volt DC to 220 AC. Plenty of trouble shooting experience.
The trailer jacks I have worked with do not need the “trailer battery” to operate. They have been wired to the “trailer plug” and are powered off the truck. The battery is just a “back up” for interior trailer lights and to power the jack if the not hooked to the tow vehicle.
To do basic trouble shooting you will need a 12 volt electric tester. A very basic one can be had for a few dollars. It nothing more than a 12 volt “light bulb” and 2 wires, a hot and a negative (ground). One can be made by using a “car” light bulb and attaching 2 wires. One to the side of the bulb, which is the “ground” and one to the bottom.
Take the voltage tester and check if there is power at the jack motor. Some jack motor only have 1 wire, the power lead. The unit is grounded to the trailer when bolted on. “Touch” the hot lead to the “hot” (power) wire, usually red on the motor and touching the ground wire to a non painted surface of the trailer or negative terminal on battery. If it lights up than there is power to the jack motor and there maybe a problem with the jack motor.
But there maybe a problem with the grounding wire. It the jack motor has only 1 wire and is “self grounding”. Take a length of wire and connect to the negative terminal on the battery and attach to an unpainted surface on the jack like a screw and see if the tester lights up. Or see if the jack works.
The majority of times there is a loose wire somewhere and or a bad ground. Wire pull out/come loose in the trailer side plug. Most tow vehicles that come with a factory installed tow package rarely have wire issues on the “towing side”.
This is a link that is usually the industry standard for trailer wiring.
https://www.google.com/search?q=trailer+wiring+harness&rlz=1C1AVNE_enUS611US612&espv=2&biw=1600&bih=775&tbm=isch&imgil=JoIgTLcQRBP0zM%3A%3B5LaCLfxhf_vkAM%3Bhttps%253A%252F%252Fwww.etrailer.com%252FWiring%252FPollak%252FPK12706.html&source=iu&pf=m&fir=JoIgTLcQRBP0zM%3A%2C5LaCLfxhf_vkAM%2C_&usg=__jQ1lo_lxZRPH253lxQR_hw_N1kM%3D&ved=0CEkQyjdqFQoTCNbczO611MgCFUkbPgodcNYLvQ&ei=3PcnVtb-Kcm2-AHwrK_oCw#imgrc=JoIgTLcQRBP0zM%3A&usg=__jQ1lo_lxZRPH253lxQR_hw_N1kM%3D
Trailer batteries depending on how they are wired should be “trickle charged” when being towed. But if lights are used a lot when the trailer is not hooked up will drain them quite a bit. The jack motor is small but requires/draws a fair bit of power when used without the truck hooked up. It the trailer is only being towed for short distances there is a good chance the battery never gets a “full” charge". Constantly “drawing down” the battery and not charging back up to “full” more times than not will greatly reduce the battery life.
Most car parts store will check the “condition” of a battery for free. Just take the battery or the trailer to a Napa, Auto Zone etc and they will perform a “load test”. It will “say” good, weak, or bad. A weak or bad battery may still show that it is “putting out” 12 volts, (most batteries charge to 13+ volts) but will not “hold” that “charge” for very long and or they have dropped in “amp” output. They will “die” very quickly when a “load” is being drawn.
A basic “multi-tester” is handy to have. They can test both DC voltage and AC. They can also do an “impedance” check on the wiring which tells you if there is a “full-strong” 12 volts being delivered or a “weak” 12 volts. You can check the ground wires, check to make sure there is not a “break” or loose wire etc. A multi-tester will let you check and see how much “power” is in the battery. A load tester can be had for $20-30. Simple to use and will give you the voltage level also.
An inexpensive “multi-tester” manual can be had that explains in layman terms how to go about various “trouble shooting” methods. Plenty of websites also.