That would be a mini-DVI to VGA adapter.
I will assume that you have disconnected and reconnected the adapter and cable and re-connected them during your attempts already. (Whether they appear to need it or not.)
From the sound of it, you may have already done all this. But, the only things I can recommend are:
1. On the TV, go into the Picture settings and reset the RGB-PC mode back to defaults.
2. After that, same place, select Resolution and make sure it is set to 1366 x 768.
3. On your Mac, use Display Config X and set it to the following, particularly the horizontal and vertical frequencies.
4. Reboot - with the TV on first and already set to the VGA input.
If all your devices are working properly, you should be good to go.
At this point, you've done everything you should have to do to get it working. If it's not, then need to troubleshoot to determine where the fault lies... MacBook port, adapter, cable, TV port. The only way to do it is try connecting the Mac to another VGA monitor and/or connecting another VGA output to the TV and switching adapter and cables out.
If you need to carry the MB to work and try it on a monitor there or friend/family...
You can also try LG's live chat support to see if they have a clue as to the cryptic error message -
link.