Can't delete Exchange account from Mail or Internet Accounts preferences
The thing is since MountainLion with cached preferences. You could try the following really hard solution (last line of defense), which trashes all you Apple Mail settings and all of your Internet accounts including twitter accounts etc. et al:


backup your Library folder to somewhere else like your Desktop or another HD
from Terminal, delete the folder "~/Library/Containers/com.apple.mail" and "~/Library/Containers/com.apple.internetaccounts"
from Terminal, delete Apple Mail Preferences with "defaults delete -app Mail" (this resets your apple mail plist to default values)
logout and login again.


If this doesn't work, copy your Library folder back and we have to find another way.