How do I enable full screen mode in Google Chrome on an iPad?

I can't explain that. It might have something to do with the website that your on at the time, but I can't reproduce the behavior on my iPad when using Chrome.

You could try clearing Chrome in Settings>Privacy. I would clear the data cache at the very least. Close Chrome completely and maybe even try force restarting the iPad and see if that fixes it. It might not help, but it will not hurt anything.

Press the Home button two times quickly. You'll see small previews of your recently used apps. Swipe left to find the app you want to close. Swipe up on the app's preview to close it

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.

I did discover that the "full-screen" on Chrome assumes the iPad is in what we call portrait mode in real-life.

Swipe up from the bottom of the web page and the toolbar disappears which then enables the full screen mode. As soon as you swipe down on the screen, the toolbar will reappear. This does not function in the same way that a true full screen mode works, like it does in iCab Mobile, but that's as good as it gets in Chrome.

why sometimes the toolbar disappears when I swipe up while many others simply doesn't? Any clue? Thanks