You can put this solution on YOUR website!
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.