Tom,
Yeah, something's definitely a bit messed up there. It seems like the _extensions table and likely the _menu table entries for the extension were not properly removed. In that case, the uninstallation process can't work because Joomla uses the XML files from the installation to know what menues and other things need to be removed from the database.
I wouldn't recommend trying to modify the database to remove that stuff manually.. yet. What I would recommend is unzipping the com_jlinked zip file and copying the contents of the /admin folder inside to /administrator/components/com_jlinked. Then, copy the jlinked.xml file to the administrator/components/com_jlinked folder as well. For good measure, if it doesn't exist, create the /components/com_jlinked folder. You don't need to move anything into it, Joomla will just delete it all on uninstallation.
Then, try to use the Joomla manager to Uninstall like you do at the end of the video. While there, also check if the JLinked Authentication, Content, System, or User plugins show as being installed. If so, try to uninstall them as well. With uninstalling any of them, you'll still likely get some errors, but that will hopefully, properly, remove the entries from the _extensions and _menu tables allowing you to re-install the component.
Let us know how that goes. Unfortunately, all the copying and moving of files isn't something we can do from just the admin access to your site. Hopefully you can do the above easily and it helps you to make progress.
If not, or you run into other issues, just let us know. We'll gladly help how we can!
Thanks,
Alex