Try this: First of all, remove any other Java versions & then reinstall Java 1.5.0_11. Use the JRE offline installer.
Archive: Download Java 2 Platform Standard Edition (J2SE) 5.0 Update 11

Next, delete LW's preferences folder when LW's completely closed. See points 2 & 3 below. If you've used older versions in the past, there could be a folder in either or both locations.
Fixes for Limewire when LW isn't operating as it should
Doing this will kill all of your settings within LW's options...you'll have to go back through & reset everything afterwards

Now reinstall LW. If you're using the latest version of LW, remember to use the offline installer (when you click to download & the box pops up, cancel that & you'll then see the option to download the offline installer).

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