Uninstall both Java using add/remove programs and Limewire using the uninstall file

Then try this Java. Download the offline installer (I have used this version for the last 10 or so versions + It fixed it for someone else having similar problems)

http://www.java.com/en/download/windows_xpi.jsp

Then install Limewire again (also using the offline installer)

www.limewire.com

Also incase Limewire won't install this time either try to turn off all antivirus/spyware and firewalls while installing. You can just disconnect from the internet if you do not feel safe doing this