I share your pain, as I used to have this same problem and in the past had written scripts to accomplish this. Today, all you need to do is use sudosh. Sudosh (an RPM) works with sudo to provide a shell that users would use for full root access, which ensures that the detailed logs are kept.
Sudosh actually records all keystrokes with the output, and can play back entire sessions like a VCR! It is very simple to set up and actually comes in two parts, sudosh and sudoshd. Sudosh is the actual program that one would call from sudo.
This was first published in April 2007