Have you tried installing the Java 1.6 beta? Here's a link & take the JRE. Make sure you uninstall your old version of Java first.
http://java.sun.com/javase/6/

After you've done that, reinstall LW using it's offline installer
http://www.limewire.com/english/content/download.shtml
(no need to uninstall first, LW will do it when you install the new version)