[morzh]

Henry

See if you can find a remedy (don't even know if such exists) that could be simply sprayed upon and left without wiping to act as detergent.
Short of that...any mechanical disturbance or even running water through could damage old cloth.

If it is do or die thing.....I would use warm water....or even cold with mild detergent and the cloth on its frame (hopefully it is not cardboard) and simply immerse it in, let the solution ru off y itself and dry in aired cool place.
Maybe when rinsing the soapy water use gentle stream of clean water from a sauce pot.