Also, Microsoft has some documentation on what to try:
The important tip they offer is:
If the last attempt did not start the external program, verify that the Database Mail External Program is located in the correct directory, and that the service account for SQL Server has permission to run the executable.
I would also check the logs. Not just the mail logs, but all the database logs and the windows logs. May have some clues in there. Also, might not hurt to check for any blocking.
The above is all just my opinion on what you should do.
As with all advice you find on a random internet forum - you shouldn't blindly follow it. Always test on a test server to see if there is negative side effects before making changes to live!
I recommend you NEVER run "random code" you found online on any system you care about UNLESS you understand and can verify the code OR you don't care if the code trashes your system.