I tested the AmiBench system and the preinstalled writing app. But on both there are the keys missing.
Next I wonder: in the settings of the A1200NG there is a section to select a keyboard. But there are only PC based keyboards. But this \ | Key, which lays left from DELETE on an Amiga Keyboard is not existing on PC keyboards. So, seems to be a problem of the Emulator.
The missing keys are: Backslash \, the pipe |, and both of the brackets ( and ) on the numberpad. All those keys are "hidden" keys on PC layout keyboards.

Maybe, Amigakit should provide an Amiga keyboard configuration?!

I'll try to create a new key mapping file in OS3.2.3 which moves 'backslash' and 'pipe' to other keys that are not so important. For this I found a tool on aminet.net.

Update:
I tried out several thing to fix that, even with an AI Pro account. But nothing worked. These keys (backslash, pipe, ( and ) on number pad) are simply not there.
But, on that original Amiga keyboard there is one key without a print (right from left shift). I could reassign it with the EditKeys tool. At least I now have a \ and a |   

That sounds interesting. Could you tell me where I find the configuration file of the A1200NG? I didn't even get any documentation with my A1200NG board  .

No idea sorry.  I know a lot about the hardware, nothing at all about the software.  Maybe check the Amiberry documentation.

I have a (real) A500+ (over 30 years old). But as I know the keyboard is not compatible (or at least the cables arn't)?!

The A500 keyboard is connected via synchronous serial, same as A2000, A3000, A4000, etc.  The A1200NG 31-way internal keyboard connection is only compatible with the A1200 scanned matrix keyboard.

Another in the "I don't have working real A1200 hardware to test if my 30 odd year old keyboard is knackered" club.

My Amiga keyboard is a brand new one  .

I have a (real) A500+ (over 30 years old). But as I know the keyboard is not compatible (or at least the cables arn't)?! 

Problem: the backslash key is not working or, at least, nothing happens when I press on that key.

If I switch layout of the keyboard, for example, to german then I see characters on screen when pressing the \ | key. But neither with US nor with GB layout. Also the "(" and the ")" keys on the numberpad aren't working.

From your test, the key is physically working.  I had this same problem a couple of days ago on the PC at work when I changed from WinUAE v4 to v6, without any changes to the emulator configuration, in the same OS3.1.4 environment.

The problem/solution was that in the WinUAE keyboard input settings, the default WinUAE keyboard had backslash 'input target' incorrectly mapped to something else.  Editing the input target to backslash resolved both backslash and pipe not working.

You may need to perform the same thing in the Amiberry configuration, it's based on WinUAE.

I'll have to check if the number pad bracket keys are a problem too, I don't typically use those.

Just wanted to clarify the bad caps are in the 1084S not the A1200.  I spent a year doing everything from replacing caps to replacing the DAC, adding caps to the supply rail, etc with no luck.

Strangely the video is only effected when the Amiga is connected via RGB.  I'm not sure why composite doesn't have the same issue, maybe because the composite sync doesn't have as much noise or is better filtered, not sure.

So anyway, I discovered that the short vertical height was actually caused by the vertical adjust pot which was broken. It looks like the section of the carbon track broke off, and it could not longer reach full range needed to expand the size to fill the screen..  grok and schematics insisted this was a 220ohm pot, but that is impossible because the remaining section (about 3/4 of the track) measures 275 ohms. After several rounds of arguing with AI and describing the broken pot it finally gave in and agreed that it's possible a 500 ohm pot was used in some boards.  Pretty funny that AI was wrong 90% of the time during this repair.  I'm tempted to upload a youtube video of the dialog which got a little heated at times as grok sent me in circles.

Anyway.... these pots are somewhat difficult to find, the original impossible, but I believe I found one at digikey which might work.

What I found after replacing the other caps in the monitor was that the loss of vertical sync (screen rolls once) on disk access went away and I just got the slight screen standard warble at the top of the screen ... something every A1200 I've seen has suffered from. I remember complaining about it in the 90s lol.

Will update in a few days when the new vertical size adjustment pot arrives.

I'm trying to save someone else from the hassle I've gone thru to resolve this issue. Really hoping this fixes the issue because I am really tried of using an LCD with horrible blocky distorted pixels. Also external PC speakers do not sound as good as the 1084s.  The 1084s just sounds amazing.