I can’t turn on Airplane Mode.

Yeah, I know it's weird. I had a similar issue with the Bluetooth setting about two weeks ago. I could not turn it off to save my soul. I just kept restarting my phone and restarting it and I was finally able to get it to work.

As much as an hate to suggest this it times, it might work - resetting all settings. When you reset all settings you have to re-enter every single setting for the device and it can take quite a bit of time to go through each setting and make sure everything is the way it should be.

IF you would rather not do that, backup, restore to factory settings and then restore the backup. Or backup, erase the device and then restore the backup. I assume that you know how to backup with iTunes or iCloud and how to restore using iTunes or erase in Settings>General>Reset>Erase all content and settings.

Reset your iOS device to factory settings - Apple Support

Restore your iPhone, iPad or iPod touch to factory settings - Apple Support

Short of restoring as new without using the backup, I don't think there is anything else to try. If you would rather not restore as new, you could make an appointment at an Apple Store and asked for their help. But their help may come to down to a clean restore as new anyway.

Try a forced restart, no data is affected by this. To force restart your device, press and hold both the Sleep/Wake and Home buttons for at least ten seconds, until you see the Apple logo.

OK, try turning WiFi off, restart the iPad and then see if you can get Airplane mode to turn on and stay on. Then turn WiFi on again and see if you can toggle Airplane mode on and off.