If you've been using Java 6 & reinstalling doesn't help, try reinstalling Java JRE 1.5._11
Java SE Downloads - Previous Release - JDK 5

Delete LW's preferences folder. See point 3 below (also point 2 if you've used much older versions of LW in the past). This will kill all of your settings within options, you'll have to go back through & reset things afterwards.
Fixes for LimeWire

Now reinstall LW & make sure to use the offline installer.
LimeWire: The Official Site for the Fastest File Sharing Program on the Planet

If that doesn't help, try changing compatibility. That's point 5 in the 'fixes' thread.

If it's still not opening, have you installed/updated any security programs lately? Make sure that LW's on the exceptions/allowed list for any antispyware etc.