Some people have had trouble opening LW if they're using Java 1.6.0_01. Is that the Java version that you have? Has this problem started since you updated Java?
Try this version instead. 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). If you haven't used older versions & only see the LimeWire folder within Application Data...just delete that one
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.

Reinstall LW & make sure to use the offline installer (when you click to download, ignore the download box that pops up. Look in the orange box on that download page...you'll see an option to use the offline installer).

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