This appears to be pretty much what I would want, except that I would like to be able to capture the machine name associated with the user who is making the connection. When I view the Activity Monitor in SSMS under the Management node, I can see the client computer name identified as "Host", but it is not clear how I can access that value from the trigger.

The reason this would be helpful is that there might be several people who share access to an account. Using the client computer name to distinguish where the login came from would help identify the person making those ddl changes.