Start by drawing a semicircle whose diameter is on the short side of the rectangle. Draw a line tangent to this semicircle from one of the far vertices, dividing the rectangle into a triangle and a quadrilateral. Draw the incircle of the triangle. Draw lines from the centre of the incircle to each of the three points where the incircle touches the triangle. Draw a line from the centre of the semicircle to the point where the tangent line touches the semicircle. Et voila, a square and four different kites.

There is a special case when the ratio of the sides of the rectangle is (1+sqrt(2))/2, when the upper and left kites are identical. We avoid this problem by drawing the semicircle on the long side instead in that case. Another special case is when the rectangle is a square, when the lower left and mid left kites are identical. This cannot be avoided by using the other side of the square, which is why the rectangle was specified in the question as non-square.

By the way, it’s a nice little fact that the largest kite, the right one in the diagram, will always be exactly half the area of the overall rectangle.