Thank you for the post.
"My question is, what is the easiest way to get FMS 13 web services running again? Re-install?"
Yes. I would uninstall and reinstall FileMaker Server 13 as this no longer even requires a restart of the machine.
If you uninstall and reinstall, then the instance of the FileMaker Web Server should be properly reconfigured. A redeployment may also work, so might try that first.
Please let me know if any of the above needs additional clarification.
Thank you! We will head down that road.