Some people have had trouble with Java 1.6.0_01. Try this version instead. Use the JRE offline installer & remove your other Java versions before you install this one.
Archive: Download Java 2 Platform Standard Edition (J2SE) 5.0 Update 11
Next, delete LW's preferences folder when LW's properly closed. See point 3 here (also point 2 if you've used older versions of LW in the past).
Fixes for LimeWire
This will kill all of your settings within options, you'll have to go back through afterwards & reset things.
Reinstall LW & make sure to use the offline installer (when you click to download LW, the download box will pop up. Cancel that & you'll see an option to download the offline installer).
If it's still not opening, try changing compatibility (point 5 in the 'fixes' thread).