Cannot delete accounts in Mail.app

Question

Title states the problem. I cannot delete accounts in Mail.app whether in the Mail preferences or Internet Accounts tab of the System Preferences. I have dead e-mail addresses and when I click the minus, in Internet Accounts, the account flashes off and comes right back, complete with a fancy little animation. When I do it in Mail, it tells me "The operation couldn't be completed." and I should try going to Internet Accounts.
How do I purge these accounts from the system? I've already deleted the mailboxes in ~/Library/Mail/V2.
Thanks.

Answer 1

Launch the Console application in any of the following ways:
☞ Enter the first few letters of its name into a Spotlight search. Select it in the results (it should be at the top.)
☞ In the Finder, select Go ▹ Utilities from the menu bar, or press the key combination shift-command-U. The application is in the folder that opens.
☞ Open LaunchPad and start typing the name.
The title of the Console window should be All Messages. If it isn't, select
          SYSTEM LOG QUERIES ▹ All Messages
from the log list on the left. If you don't see that list, select
          View ▹ Show Log List
from the menu bar at the top of the screen.
Click the Clear Display icon in the toolbar. Then take the action that isn't working the way you expect. Select any lines that appear in the Console window. Copy them to the Clipboard by pressing the key combination command-C. Paste into a reply to this message by pressing command-V.
The log contains a vast amount of information, almost all of which is irrelevant to solving any particular problem. When posting a log extract, be selective. A few dozen lines are almost always more than enough.
Please don't indiscriminately dump thousands of lines from the log into this discussion.
Please don't post screenshots of log messages—post the text.
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