Kindly note that updates are discovered, downloaded, and applied by Joomla itself. It's not our code and we have no say in its development.
I would recommend following the instructions in https://www.akeeba.com/documentation/admin-tools-joomla/update.html#update-troubleshooting.
If all else fails, remove the Download Key from the update site and Save & Close. Come back to our site, log in, and go to https://www.akeeba.com/download/official/add-on-dlid.html. Copy a Download ID without any whitespace (spaces, tabs, newlines, etc) around it. This is important. Joomla will stupidly not warn you if you are pasting a Download Key with whitespace characters, even though whitespace characters break the download. Go back to your site's System, Update Sites, edit the Admin Tools update site, and paste the Download ID you copied. Click on Save & Close. Go to System, Updates, click on Refresh Update Information (or whatever they have changed its translation string this week) button at the top of the page. Now you will be able to update Admin Tools.
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!