While I was thinking about this problem, I got an idea: what if the board wasn't flat, but instead spherical? Or cube-shaped? After some trial and error, I was able to come up with answers, and I was able to find one cube-shaped board that seemed to work. Here is the design of the cube:
If you've ever folded a cube, you know how these designs can get turned into cubes. The folds should go between c & d, between f & g, between i & j, between 7 & 6, and between 4 & 3.
Here is how the pieces move:
All the kings attack all squares that are adjacent to them.
As an example, the king on c5 attacks the squares b6, b5, b4, c6, c4, d6, d5, and d4. The fold between c and d doesn't hinder the movement.
Another example: the king on a5 attacks the squares a6, a4, b6, b5, b4, l6, l5, and l4. The squares in column l become accessible after folding the cube.
The movements of the knights is like this: they move two steps in one direction, and then 1 step in a perpendicular direction.
As an example, the knight on e6 attacks the following squares: d8, f8, f7, g5, f4, d4, c5, and d7. The squares f7 and d7 are the least obvious. To reach f7, the knight first moves two steps to the right, to g6. Then it moves up to f7. It attacks 2 knights and 2 kings: the knights are on d8 and f8. The kings are on c5 and g5.
Another example: the knight on e7 attacks the following squares: d9, f9, f7, g5, f4, d4, c5, and d7. It attacks 2 knights and 2 kings. The knights are on d8 and f8. The kings are on g5 and c5.