Try the advice in points 1, 3 & 5 here
Fixes for LimeWire
When you reinstall Java, make sure to use the JRE offline installer. Remove your old Java version before you install the new one.
If none of those help, make sure that LW's on the list of exceptions/allowed apps for any antispyware programs you're using. Some antispyware will see LW as a threat (it's not) & prevent it from opening.