Firstly, while I'm clicking on logout button, its showing "Oops! the link appears to be broken".
Secondly, once I use this blanket url in a browser, the next time if I open the administrator link directly, without using this blanket url, the administrator page is opening directly.
First: If you're getting an error on logout - it's because you have an invalid redirect URL in the plugin settings.
Second: if you read the description, you would know that this is how the plugin behaves:
"Once you've clicked the second link, your AdminExile key will be active until your session expires (or until you close your browser)."