I will save you the trouble of wading through xda threads (but there is good info there that's still worth a read) for those who are at&t customers in good standing and want the non-hacker-style method.
If you have the 10o software version you can't get the JB update at this time (I hope they fix that because a lot of us are still at 10o, having missed the last chance at the 11c update over Easter weekend). So you need to get 11c before you can get 20j (JB).
There are many ways to get 11c but the least risky way in my opinion is to get AT&T to push it to your phone. I was surprised that their L1 people would have any way to do this, but they do. I asked them in a chat but you could call them also. Just tell them you want to get the JB update but you understand you need software version 11c first, but that update failed, and ask could they push 11c to your phone. Many people have done this successfully. I actually had two people try to push the update to me and neither gave me the slightest bit of grief over it (the update was failing due to a change I had made, which I found out about later).
Once you have 11c you should shortly get the message for the JB update and if you don't, you can check for the update. If you want to check more frequently than every 24 hours, go to Settings -> General -> Applications -> ALL -> com.lge.omadmclient and clear data. That will let you check again immediately, though there's no point in doing it frequently.
Some people get an "authentication error" when trying to install the OTA. This may be due to an error on AT&T's end - so if you get that, try contacting them.
The 11c update is about 18 megs and the 20j is 366 so you'll probably want to be on wifi for that one.
For the other methods, if needed, definitely go to xda.