The reason that the P2 controls don't work is because the Dip Switch setting is set to "Upright" instead of "Cocktail".
A DK upright cab has a single joystick and jump button -- during a 2-player game, players would take turns using the one stick and button.

A DK cocktail has a stick and button on each side (note the instruction sticker above the monitor) -- during a 2-player game, the screen orientation and controls would alternate between the players sitting on opposite sides.

If you want Player 2 to be able to play from their joystick in "Upright" mode, you can map the controls to accept inputs from both sticks/buttons at all times. (See EssexMame's post above for the down-side of this approach.)
To do that for "P1 Up":
* Highlight "P1 Up" in the "Input (this Game)" menu
* Press Enter
* Press Escape (sets input to "none")
* Press Enter
* Press P1 joystick up (sets input to "Up")
* Press Enter
* Press P2 joystick up (sets input to "Up or R")
Repeat this process for the rest of the P1 inputs.
If you make a mistake or want to go back to the default setting:
* Press Enter
* Press Escape (sets input to "none")
* Press Enter
* Press Escape (returns input to default setting)
In closing, please tell me you aren't thinking about playing . . . Arkanoid . . . with a joystick??!!??

Scott