If your login check DOES work, I might have your solution.
When I activated the plugin, it didn’t create the database tables (didn’t run the install function). I don’t know why this happened.
Try going to https://example.com/wp-admin/options-general.php?page=twitter-tools.php&activate=true (example.com being your WordPress installation URL). This should run the activation script and create the table wp_ak_twitter. After that you can click “Update Tweets” and you should see your tweets.