I assume you mean you have 10.4.11, because there's no way software update would say everything is up to date with 10.4.1.
That being said, there is no Java 6 (or 1.6) for Mac OS X 10.4. Apple never released one. Which is actually a blessing in disguise, because Limewire is a great way to have your system overrun with unwanted garbage not to mention find yourself being sued over something you downloaded. Just because you have a Mac does NOT mean you are immune to any and all security threats out there.
So, my strong advice to you would be to simply delete Limewire, and any other file sharing program(s) you may have installed, and just leave it at that. If people stop illegally downloading copyrighted stuff, and they still don't buy it, then the entertainment industries of the world will be forced to consider the possibility that maybe it's time to stop pumping out (barely) warmed over versions of the same crap over and over again. That or they'll go out of business like they probably should have about 10 years ago or more.