Remember, the problem is with the database server. Uninstalling Admin Tools will NOT fix the database server. It will only lose all of your Admin Tools settings. You will make things worse, not better.
You can simply (try to) drop the table, then visit Components, Admin Tools again. It will detect the table is missing and create it afresh. The only thing you miss is the PHP File Change Scanner status regarding files. Which means that you just need to run it again and mark all files as safe.
The problem is dropping the table. The reason I told you that you might need root access is that most of the times I had this issue I was not able to repair or drop the table. MySQL insisted the table isn't there. I had to drop the table by deleting its information from the MySQL schema (DANGEROUS! That's why it requires root access.), stop the MySQL server, remove the table files from MySQL's data directory (assuming it's a MyISAM table, no idea what to do for an InnoDB table) and then create it afresh. What was much easier for me to do was to either restore a backup or -on dev servers where backups are not a given- export the entire database minus that table with phpMyAdmin, create a new database, restore it there and edit Joomla's configuration.php to point it to the new database.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!