Get a multimeter if you don't have one. (Inexpensive is fine...most tests are yes/no tests. You have voltage or you don't. You have continuity or you don't.)
Set it on DC Volts.
Use the probes to test the voltage in the socket. The center should be positive (red) and the barrel on the side should be negative (black). Make sure the probes are making good contact with both socket contacts. A little caution is in order, because if you short the socket with the probes, it may blow the fuse....but then you'll know the socket works.
If you read 12 volts, the issue could be the connection between the 12-volt TV plug and the socket. These 12-volt cigarette-lighter style sockets are not the best idea. In automotive uses, there are many situations where the plug and socket don't mate properly, and the power is intermittent or just not there...even though the socket is live. I have an excellent USB charger in my truck that must be coaxed to make a solid connection. It has a pilot light that betrays it loosing contact, and I have to reseat it somewhat regularly. Sometimes it's necessary to strengthen the bond between the negative pole on the plug and the negative pole in the socket (barrel) by bending the tang on the plug. If the negative pole on the plug is a kind of spring-loaded button, some
electronics cleaner may improve the connection.
Since this is just a simple socket, failure is possible but unlikely.
Of course the socket can be replaced, but I'd exhaust all other options first. If you can remove the socket easily, do so and examine the wires connecting to the socket. Maybe one of the wires has a "cold" (bad) solder joint. Hitting the connections with a soldering iron may do the trick. If nothing else, you'll know how to replace it. Sometimes just being a bit ham-fisted and pushing hard while twisting the plug does the job.
About that fuse....so, the fact is, many people struggle to get a true reading on a blown 12 volt spade (automotive-type) fuse. The best way to read the fuse is to pull it and shine a flashlight through it from behind. I once solved a camp neighbor's electrical problem that way. He was absolutely sure the fuse was fine, but it wasn't. Shining the flashlight from behind is the most reliable way to verify that it's not blown. An alternative is a continuity test (ohms tests) with the multimeter.
P.S. on the fuse...check them all. My camp neighbor was also absolutely certain that his failed circuit was on a particular fuse. It was not. It was on another fuse. Check all of them. It takes only about 60 seconds to do the whole panel.
This system is super simple. Aside from the mystery fuse in the TV plug that others mentioned and you checked, you can see from the e-Trailer items that some sockets also come with a fuse...nothing like triple redundancy!
Perhaps there is another in-line fuse that failed.
In the end, if the fuse in the panel is good and the socket is dead, and if it's possible to do this easily, cut the wires that connect to the socket in such as way that you can reconnect them easily with a
crimp tool. It would be very unfortunate, but it's possible that the wire between the panel and the socket is damaged. Use the multimeter to test the voltage on the wires feeding the socket. Before you perform this test, pull the fuse on the circuit so you don't cause a short when you cut the wires. Replace the fuse, perform the test, and then pull the fuse before you splice the wires back to the socket.
In my rig, I had a screw driven thru the floor into the fill pipe for my fresh tank. I also found a cabinet screw driven straight into a 120 volt wire...then backed out again and left there! Don't rule out anything, because this is the simplest of circuits and having the socket fail is possible but very unlikely.
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