Installing Limewire (or more important installing Java) may require some administrative privileges to properly register the VM or so that the startup program will get these privileges that allow integrating the support for Magnet links in your browsers (also need for installing the Java VM to get some hardware support).

The administrative privileges are also required to allow the installer to insert the registry keys that allow uninstalling the program or upgrading it later.

Once Limewire is installed with such administrative user account with privileges, you can run it safely from any user account (including a guest account).

So logon with a administrative "power" account, install Limewire from here (you don't need to run it immediately with this account).
Then logoff and logon with a limited account, and LimeWire (and Java) will run correctly for all user accounts on your system.

Make sure that you can effectively browse the Java demonstration pages on the Sun Java download web site in your browser. If this does not work there, there's nearly no chance that Limewire will start correctly.