Washing machines often have pumps built in. Could you use the inbuilt pump to supply the water, instead of just draining it?
I remember an old washing machine my mum had used to have a "suds return" cycle, where it would pump out the washing water into the laundry trough, so that you could use it again by pumping it back in on the next cycle.
You could program the arduino to allow for that, to make getting the washing out easier.