Thank you for posting.
Which version of Java is installed on the machine? An incompatible version could cause an issue with the web publishing. The latest FileMaker Server 11v2 version should be compatible with Java Update 20.
Because the Web Publishing was working at one time, it seems likely that the cause of the error is a corrupt installation or an incompatible software update.
It is curious that the web publishing still functions despite the error. This would indicate perhaps that it is trying to start the publishing engine twice, but failing on one attempt.
Did you try "Edit Server Deployment" to redeploy?
A reinstallation of the latest FileMaker Server version is likely to resolve the Admin Console error. When uninstalling, please be sure to backup your Data folder, completely remove the FileMaker Server installation directory, and lastly uninstall Java.
The errors you mentioned from the logs seem to be general HTTP errors:
400: Bad Request
404: Not Found
Do these seem to have occured at the time the issue started?
Thanks! The reinstall in under way. Will post results once completed.