What version of Java are you using? Try this version. Use the JRE offline installer & remove your current Java version 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).
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.

Now 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).