I forgot a part of the link here is the right one:
http://java.sun.com/j2se/1.4/download.html
You should look up the row of your operating system on the left and then use the DOWNLOAD link on the left ('JRE' column).

the above is not LimeWire it is just what you need to run Java Applications. It is called a Java Runtime Environment (JRE). This is basically a 'layer' between the Java application and the hardware of your computer. usual programs don't need that 'layer' but they are dependent on one operating system. With Java you have a 'layer' for all operating systems and that way you can run any java program on any java-able machine. this is why this 'layer' is often called virtual machine - it simulates a computer (about like that). enough of this.
You still need to download LimeWire and install it. The default LimeWire comes with a small virtual machine but if a JRE is installed on your system it will use it instead because it is faster.

that's why I said you might as well try to uninstall and reinstall Limewire first. most of the time something just went wrong during installation or the download...
And this would solve the problem much faster. The JRE is about 50MB large so downloading might take a while...

good luck.