Which version of Java are you using? If you haven't already, try this version. Use the JRE offline installer & remove any other Java versions before you install this one.
Archive: Download Java 2 Platform Standard Edition (J2SE) 5.0 Update 11

If that doesn't help, 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 (there could be a folder in either location). This should help with the old splash screen problem as well
Fixes for LimeWire
This will kill all of your settings within options, you'll have to go back through afterwards & reset things. If you're not sure of your settings, go through LW's tools > options & write them all down, so that you can reset things later.

If it's still not opening, try changing compatibility (point 5 in the 'fixes' thread).

If nothing works, reinstall LW & make sure to use the offline installer (when you click to download LW & the dialog box pops up, cancel it & you'll see an option to download the offline installer).