You're experiencing the WordPress' White Screen of Death, if you Google it, you'll see lots of explanations.
Sounds like you have some bad code in a plugin. If you have backups, you can try re-installing WP and your DB from backups. Otherwise you'll need to disable your plugins. You can do this by going to wp-content/plugins on your server and renaming all of the folders temporarily (put an x or something in the folder name). Hit your admin url and all of your plugins should deactivate. You can then rename the folders and log in to admin and slowly reactivate plugins until you isolate the plugin that is causing the problem. If you white screen again, just rename the last plugin folder for the plugin you reactivated just before the white screen and it should isolate the plugin that is causing the problem.
My guess is it is the 'badge' plugin (obviously).
With the plugin off you should then be able to remove the widget.