Nicholas (Akeeba developer) already told us what to do in such a case:
I get a blank page or Internal 500 error accessing Akeeba Backup or trying to backup
"If you are receiving a blank page or a fatal error when trying to access Akeeba Backup and/or receive an AJAX error during backup right after upgrading to Akeeba Backup 3.5.x, please note that this is the result of a partial upgrade. There is a known bug in the Joomla! extensions installer (which is also used by our Live Update to perform the actual update). When upgrading to a version which has a different count of top-level directories, Joomla!'s extension installer "forgets" to copy all files and folders or remove the old ones. This usually happens when upgrading from Core to Professional and when upgrading between minor releases, e.g. 3.4 to 3.5."
Quite logical: DO NOT UNINSTALL OR DOWNGRADE Akeeba Backup!
Instead, UPGRADE IT:
"The solution is usually very simple. First DO NOT UNINSTALL Akeeba Backup and DO NOT DOWNGRADE to an earlier release. The former will remove your backups and settings. The latter will cause Akeeba Backup to fail. Instead, go to our Downloads page and download the latest installation package of Akeeba Backup. Go to your site's back-end. Go to Extensions, Manage Plugins and make sure that the "System - System Restore Points"(*) plugin is either not installed or, if installed, is not published. Then go to Extensions, Manage Extensions and use the Browse button to locate the installation package you downloaded. Then click on "Upload and Install". You may have to repeat the installation a second time."
Hope this helps.