Please help, I've tried but just ended up coming up with about ten equations.

Let d be the distance from the diagonals' intersection and the larger side, so the distance from this point to the smaller side is d + 4 cm. Then d is half of the rectangle's shorter side (call it x), and d + 4 cm is half of the longer side (call it y).

The perimeter of the rectangle is 56 cm, which means

2x + 2y = 56 cm

or

x + y = 28 cm

In terms of d, this becomes

2d + 2 (d + 4 cm) = 28 cm

Solve for d :

2d + 2d + 8 cm = 28 cm

4d = 20 cm

d = 5 cm

Then the sides of the rectangle have length

x = 2d = 10 cm

y = 2 (d + 4 cm) = 2 (13 cm) = 26 cm