What version of Java are you using? You can either try updating to the latest version here (I have this one & LW's working fine): Java SE Downloads
Or, if that latest version gives you trouble, try this older version, which has worked well for lots of people: Archive: Download Java 2 Platform Standard Edition (J2SE) 5.0 Update 11
Whichever Java version you use, download the JRE offline installer & remove any other Java versions before installing another one.

Now download LW again & make sure to use the offline installer. When you click to download LW, cancel the dialog box that pops up & use the option to download the offline installer.

If you're still having problems after that, try deleting LW's preferences folder. See point 3 below & also point 2 if you've used older versions of LW in the past (there could be a folder in either location).
Fixes for LimeWire
This will change all of LW's settings within options, your old preferences from previous LW won't be maintained. Once you've reinstalled LW, you'll have to go through options & reset everything.
In case you're using Vista: LimeWire Pref Folder in Vista