I finally resolved this issue, and I wanted to share the information in case anyone had this same problem.

Before LimeWire would load upto the "Loading Download Window" then the window would disappear! I cannot tell you how many times I uninstalled and reinstalled different versions of Java and then LimeWire. 4.12. After some work of finding out what software was not compatible with LimeWire, I found that the software that came with my router was interfering. (I have a D-Link router that came with Network Magic software.) Believe it or not, after I uninstalled Network Magic, I have had NO problems loading LimeWire.

So, I just wanted to pass this info along. Have a good night!