Just enabling the module isn't enough.
Since Kohana 3.3 the autoloader is case sensitive, so MODPATH/mysqli/classes/kohana/database/mysqli.php
should become MODPATH/mysqli/classes/Kohana/Database/MySQLi.php
for example.
You have to edit your APPPATH/config/database.php and set the driver to 'MySQLi' or set Database::$default to the correct default database instance name. (The one using 'MySQLi' as the driver.)
And just to be sure, do NOT edit config files outside of APPPATH. Instead, copy the file to APPPATH and edit that file. You generally should not have to edit files outside of APPPATH. Updating modules to work with a new version is allowed of course. But config files outside of APPPATH should serve as examples/blueprints.