Hi smaker.
welcome to the Apple (user-to-user) Support Community!
Re: "Backlight on magic keyboard" "The manual adjustment on keyboard back lighting does not work."
Am no expert in this area, yet your questions deserve some response.
Wondering if this is the manual setting you have been using:
"If you want to manually adjust the brightness of the keys on your Magic Keyboard,
go to Settings > General > Keyboard > Hardware Keyboard."
(Above is from: Set up and use Magic Keyboard for iPad - Apple Support)
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Re: "When I increase the brightness of the screen, the back light goes off".
You might want to try restarting your iPad (Eg: Deletes temporary files, re-establishes connections, may update recent setting changes, and often restores order if something is not working as expected.)
If needed: See how to: Restart your iPad - Apple Support
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re: "It would be nice if the accessibility option included, turning on the back lighting all the time."
If the issue continues, you may want to send feedback to reach the Accessibility Support Team:
They will likely not reply, yet your feedback will reach those whoi could make changes if others also experience similar situations.
To provide feedback about an accessibility feature on an Apple product,
email the Accessibility team: accessibility@apple.com.
All the best :-)