Local private network issue.

It sounds like you are having a difficult time with Windows networking. Let's break this down and see if we can find a solution.
Firstly, it's crucial to ensure that all items are configured correctly. Generally, these are the options we need to confirm are set properly:
- Confirm that all computers are in the same workgroup.
- Make sure Network Discovery is turned on.
- Confirm that file and print sharing is turned on.
- Make sure you're on a 'Private network'. If you're on public, the default settings might prevent sharing.
- Check the status of the 'Function Discovery Resource Publication' and 'Function Discovery Provider Host' services – they should be set to Automatic.
- Turn off password protected sharing.
- Make sure the correct IP range is set if you use Windows firewall - it should align with your router's DHCP IP range.

After verifying these settings, if you still can't connect to the shared folder from the laptop, I would recommend trying a few troubleshooting steps:
1. Try to temporarily disable any antivirus software and firewall on both computers.
2. Try to access the shared folder by typing \\IP address\SharedFolderName in the Windows Explorer address bar. Replace "IP address" with the IP address of the desktop and "SharedFolderName" with the name of the shared folder.
3. You mentioned that the laptop can't ping the desktop by IP. Did you check if laptop's firewall settings allow ICMP Echo Requests? If not, it could be the reason blocking the ping.
4. Try resetting the TCP/IP Stack of both computers. Open command prompt as administrator, type "netsh int ip reset", then restart the computer.
5. Try to use the hosts file to map the desktop’s name to its IP address.

Always verify that you have made changes on both the laptop and the desktop, as the settings need to mirror each other for successful file sharing. Please do let us know how things go.