A big
thanks to anyone who attempted to help me out -- especially Birdy.
I'm not sure exactly how, but I'm up and running again.
As any forum is only as good as the people who post and, more importantly, their willingness to follow through
to the very end -- including a description of any resolutions they were fortunate enough to obtain -- I'll attempt to [completely] describe what happened:
1. Tried disconnecting and reconnecting after a 5 minute delay --
No Luck.
2. Read up on the Win XP SP2 Patches (both the Microshaft version and the Lvllord version) --
Didn't like what I was reading from others who tried them.
3. Deleted the
.limewire [yes, DOT limewire] folder from the
Documents and Settings folder on my boot partition. Upon, starting LimeWire (LW) again, however, this file was recreated. I suppose that means that you're essentially refreshing this file. Anyway --
This didn't work either.
At this point, I started to study Birdy's words a bit more closely. In particular, the text relating to installing a new version of LW caught my attention.
4. Used Windows Control Panel to remove my old version of LW 4.8.1 Pro.
5. Reconfigured my Linksys WRT54GS router, by entering
192.168.1.1 as a URL, so that the port forwarding (under
Applications & Gaming/Port Range Forward) I previously created was
disabled. I also took this opportunity to ensure that the
UPnP feature (under
Administration/Management) was
enabled.
6. Downloaded the Basic version of LW 4.12.11 -- with the
Offline Installer [as I always opt for].
7. Opened up the Windows Firewall utility and deleted the LimeWire exception.
8. Closed all open applications.
9. Used
Start | Run to install the Basic version of LW 4.12.11.
10. Ran LW --
still no immediate connection.
10. As I noticed that I wasn't asked by the Windows Firewall program to
Unblock the LimeWire .exe during its execution, I checked the exceptions tab once more --
LimeWire was listed.
11. Restarted Windows XP.
12. Started LW Basic 4.12.11 --
Voilą! Also known as:
5 Green Bars! In short, it would seem that my repeated attempts to use my old version of LW -- 4.8.1 Pro -- were meeting with some sort of incompatibility issue (or issues). Regardless, I'm now able to use the Direct Connect feature in the application to transfer large work files, so, as the Pythons would say, "...and there was much rejoicing." 
By the way, Birdy. Considering that I went through the Port Forwarding procedure twice, do I need to go back and make follow-up changes now that I'm using UPnP? Although I certainly learned a thing or two by way of this experience, I don't know a whole lot about router security. Consequently, I don't want to create an opportunity for a hacker attack without knowing it. In other words, I wouldn't want to have an unnecessary open port. 
Many thanks!