If you can't load the test page then something's blocking LW's listening port. It could be an ISP or it could be a firewall within your apartment complex's setup. You could try changing LW's listening port (tools > options > advanced > firewall config) to something in the range of 49152-65535. Try port 64049. If that's no good then try some other ports within that range.

Otherwise you might have to speak to the complex's manager, or whoever is in charge of the internet there, & ask if they'll open a port for you so that LW can connect.

Did you update your Java version before LW stopped opening? Try removing your current Java version, then download the offline installer for Java JRE 1.5 update 11.
Java SE Downloads - Previous Release - JDK 5

Else, try the fixes in points 2, 3 & 5 of this thread & see if LW opens after that!
Fixes for LimeWire