Something to keep in mind ... the connection making a change, has to have access to the audit table. That and unless you want to create an audit table per database, it's usually best to use a central table per server.

Check the attached for a script that will push the trigger to all databases. The trigger is more or less the same, but also catches the application. Change the @LoggingDBName to whatever you use for your central DBA/Logging/Admin database.

As stated though, make sure your logins have write access to the changelog table.