Try options here
Fixes for Limewire. Problems with LW opening are generally Java related. I see you are using Java 1.6. Options I can see are removing all versions of Java & reinstalling 1.6, but I'd suggest installing Java 1.5 instead first. Some users have had problems with Java 1.6. You can get Java 1.5 here:
Java 1.5 (take the JRE) which is 5th on list. I would also recommend you download the "Offline" installer which means you download the installer & then install. Not install over the internet.