Are you using Vista or later? If you have not set java to run as administrator, then search for java.exe and set it to run as admin. This seems to make a big difference for connecting.

Paste %appdata%/LimeWire into the Run program or search window. Is there a gnutella file existent?

Try the connection fix again.

Ignore the you are not connected to network message on main window. The front window simply connected to the LW site in past which no longer exists. It previously gave messages from the LW site including updates.
Did you mean LPE 5.6.2 or one of the earlier betas 5.6.0 or 5.6.1?