Do you mean that you've updated/reinstalled Java using the JRE offline installer?

I'm not sure what fixed the problem for the desktop ( either deleting prefs or reinstalling/updating Java?) If reinstalling Java worked for the desktop but didn't help with the laptop, try deleting preferences folder on the laptop when LW's properly closed. See point 3 here & 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 your settings within options, you'll have to go back through afterwards & reset things. If you're not sure of your settings, go through LW's tools > options & write them all down, so that you can reset things later.

You'll know if the prefs folder has been deleted properly... when you start up LW, all settings (within tools > options) will have reverted to default.