This was one of the most informative posts on this Homegroup problem so I'm reporting that after 12+ hours effort I am finally SUCCESSFUL at resolving it.
That was the good news. The actual details that tipped it into working mode are however a little sketchy as I was trying so many things.
My best guess as to what was the ultimate factor is either one or a combination of:
- Changing HomeGroup Provider to "Local Service" Homegroup Listener stay as 'local system account' (I'd previously fiddled with a number of these following another post)
- Changing the name of the Computer even though there was no apparent name conflict on my network
- Removing old version of Java (probably coincidental)
- Turning off and on Microsoft Security Essentials
- Removing the fixed IP address my Desktop was using
Some other comments after my multiple failed attempts.
- Installing a WIFI card on my Desktop probably helped but didn't immediately resolve Homegroup issues when it was initially installed
- Attempting to use Homegroup to Share my "Drives" library was maybe one of the things that stuffed things up. When attempting to do this resulted in errors saying that root of drives can't be shared.
- Changing the Log On account for the various services was probably not a smart thing to do. I ended up in a rabbit hole of incompatible logon error messages
- Resetting the router to a basic environment didn't immediately resolve anything but may have helped the network troubleshooter fix things automatically
Good luck if you're still seeking resolutions though.