Twanneman try to disable other plugins and see if this problem resolves. I guess that there are other plugins that conflict with Jetpack and Admin Menu Editor, because if you install fresh WordPress with Jetpack and Admin Menu Editor there will be no conflicts.
I have not found anything helpful yet.
Thank you for your reply!
Actually, on my website, when I deactivate your plugin, Jetpack starts working.
But on a new clean site when I activate both your plugin and Jetpack, Jetpack works fine.
So there is no conflict between Jetpack and your plugin. There are other reasons that I have not found.
We can mark this issue as resolved.