Now it works.

I had to uninstall Java again, use CCleaner to remove any trace of Java and Limewire in the registry (remember to restart in between), and finally reinstall first Java and then Limewire.

The reason why I earlier couldn't execute the installation file, was of course because this was also a java-based program.

PS. More useful tools like CCleaner at Piriform - The best software is free!.