First, assume we're dealing with an 'ideal' situation in which all other forces are negligible (like the weights of the rope and seat). Second, can we assume Kathy is pushing perpendicular to the 24.3 degree angle? That is, is he sitting straight and is she pushing direclty against his back? If so, the free body diagram for this problem is fairly straightforward as shown in the attachment.

F (the force with which Kathy is pushing) must be large enough to cancel the vertical force W (the weight of John).

You can use right-triangle trig to figure out then numbers, including the answer to your question. This solution will give you the horizontal component of the force necessary to keep him in position.