Android - Is there any way to manually bring up the keyboard?

As I see an alternative keyboard may solve your issue, and this seems to be an acceptable solution, and you even mention something you cannot find -- hereby I proudly present:

Hacker's Keyboard

Checking its Guide, there's in fact a section suggesting such a feature:

You can configure the gesture used for this in the keyboard's "Gesture and key actions" settings menu. If the swipe action is too hard to trigger, try binding the "close keyboard" action to the "Volume down" hardware key. This won't interfere with normal use of the volume button while the keyboard is closed.

Though this applies to "close": If there are gestures you can configure, it might as well be able to have one for "open". The FAQ may reveal some more clues.

SaintWacko notes in the comments:

To be able to open it anywhere, you go into the settings for the keyboard and check the box for 'permanent notification'. It will then keep an entry in the notifications which you can tap to bring up the keyboard at any point.

So this would be an alternative possibility -- the hook being it is hooked to this special keyboard.

I had that same problem and bought a Bluetooth keyboard in order to get around it.

My math lab problem solved with the hackers keyboard! Download the hackers keyboard, then go into settings and enable ''use permanent notification''. This puts the hackers keyboard logo at the top right of your screen next to the clock and settings button. Click on the settings bar or clock at the bottom right and press ''Show Hackers Keyboard''. It will popup for you in flash and other times when it doesn't do it automatically.