Question 8853
 I think you mean AB = CD = 5,   AD= BC = 12.

 Let AC  and BD intersect at O, we see that AO=CO= BO=DO .
 [Set it be r]
 
 The circle centered at O with radius r passing through A,B,C & D.
 This is the unique circle (in the same plane of the rectangle)
 have two diameters AC & BD  whose endpoints are vertices A,B,C & D.

 By Pyth. theorem AC^2 = AB^2 + BC^2 = 169,so AC = 13 cm (=BD).
 Hence, the sum the diameters = 26 cm
 
 Since there are infinitely many diamaters in circle, the question
 should not ask "what is the sum of the diameters of all of the circles?"
 in this unreasonable way.

 Kenny