1. Double check this Setting up Java 1.5
2. If still problems, download the Java 1.5 again from apple getting the "Offline" standalone java installer.
3. Delete the LW prefs folder found here: /Users/your_username/Library/Preferences/LimeWire
4. Check your graphic settings.
5. Make sure you have plenty of ram. 500+ MB is best.
6. Do these maintenance routines: SAFE Boot & Repair Permissions
7. Try step 8 here Can't launch LW: Bouncing in Dock (click on link)

Thanks for the help. Unfortunately:
I have the latest Java runtime (it won't update)
There are no LW preferences to be deleted (I already deleted them, I guess that's why I can't find any Limewire folder under the preferences in the Library)
Graphic settings?
Plenty of RAM, I have 1GB, and 600 free right now.
I repaired the permissions, I even booted from the CD and repaired my system disk.

I had removed Limewire completely. However, upon lauching the 4.12 install, it told me that a newer version was installed. I had trashed Limewire and everything called Lime*.*, and emptied the trash.
I don't remember what I did, but next thing I know is that the 4.12 install wouldn't run anymore at some point (quickly appearing then fading out of the dock). So I installed the 4.13 which worked as always... and failed just like 4.12:

My Limewire Setup Window

My Java Preferences

My Java Plugin Preferences

And.... drum roll..... Azureus works perfectly, as it has always been for 2 months.

Looks like I'll have to turn my mom into a pirate :-)