This works well for me (I use ‘sleep cycle’) but doesn’t solve my inability to actually get tired and go to bed at night.

Have you tried avoiding screens an hour or two before going to bed? Fill that time with reading or listening to podcasts.

Does the Kindle still count? I heard that it is different from computer screens because the light is backlit, not directly shown into your eyes.

A Kindle should be fine, preferably without backlight however.

podcasts are awesome - we're kinda in a golden age for it.

How does an app predict your sleep cycle? Do you have the keep phone on you?

It tracks your sound or movement (either via the microphone or the accelerometer) as you sleep to determine your sleep cycles. You set a baseline of typical sleep for 5 days, and after that the app can predict when to wake you up.