Well, we seem to have come to the end of the road as far as possible fixes tried. The Link Trouble gave you should have gotten the Homegroups set up without any issues.
So I will still concentrate on the laptop. I still think you have a basic network problem.
When I click on 'see full map' on the desktop it shows the desktop - router - internet map (same as on the basic network information screen) but when I do the same on the laptop it says 'windows cannot discover any computer or device' despite it showing laptop - router - internet on the basic network information screen. Internet works perfectly well on the laptop and the desktop.
So, in an effort to discover some new information that may be hiding, I will ask for a couple of things from your laptop.
Disconnect the system from the internet and just leave the local network in place. Then disable ANY anti-virus programs you have running.
Next, open the Network Connections dialog (ncpa.cpl) and right click and properties for your network adapter-Local Area Connection.
Open the Local Area Connection Properties and take a picture of that. Also open the IPv4 and IPv6 properties and do the same thing. Use the snipping tool to take a picture and attach using the paperclip in the Advanced Replies window for any suggested pictures. (see Attachment)
As long as you are on a roll, take a picture of your Network and Sharing Center, and edit out personal info with Paint, then attach.
Next, on the laptop, run the Homegroup trouble shooter again, and make sure to run it in the Advanced mode and allow it to fix everything. When it finishes, you should have some type of window like the one attached. I had shut down the
HomeGroup Provider for testing. Shutting down the service also kept the Network Map from being presented.
I realize you have done these things several times. But either something is hiding in your settings, or something has gotten stuck in the registry stopping the homegroups from working. I will continue to look to see if I can find a way to reset the entire system on your laptop.
Hopefully, something will show up to help.