Reinstall this version of Java & see if that helps. Use the JRE offline installer & remove your current version of Java before you install this one!
Java SE Downloads - Previous Release - JDK 5

Some people have had connection issues due to a recent Windows update. Check out this thread & see what Mickjapa108 says about it. He tells you how to check if you have this uupdate.
LimeWire will not connect for me anymore
If you have this update & have your router set to use UPnP then let us know!

Try deleting LW's preferences folder when LW's properly closed. See point 3 below (also point 2 if you'ver used much older versions of LW in the past...there could be a folder in both locations).
Fixes for LimeWire
This will kill all of your settings within options, you'll have to go back through afterwards & reset things.