Database Error with Chat Plugin resulting in a bug in the admin area?

I installed the latest Chat plugin and suddenly when I activate it, I am not able to add any link in any post anymore. The input fields and checkboxes simply are unresponsive. I had a look at the server log and found there are several db errors. Even I can't think why this should have an effect on the general post edit field, the chat plugin is the one that breaks it.
Here is one of the errors I see in the log:

FastCGI: server "/var/run/hcgi/400458" stderr: WordPress-Database-Error Duplicate key name 'last_polled' f\xc3\xbcr Abfrage ALTER TABLE wp_wpmudev_chat_users ADD KEYlast_polled(last_polled) von activate_plugin, do_action('activate_wordpress-chat/wordpress-chat.php'), call_user_func_array, WPMUDEV_Chat->install, dbDelta, referer: