I feel for you, RF. It's hard not to make harsh judgements of a person that behaves like this.
I would now approach it by asking questions. Why does he continue to not wash dishes? Did he have to wash too many dishes growing up? Was it some kind of punishment, something used to make him feel inferior? Or, does he simply perceive it to be something that's beneath him? Maybe he could be induced to chip in some other way; perhaps taking out all of the garbage, or buying dinner once a week. But either way, he should be carrying through with the agreement he made, and be responsible.
If that doesn't work, I would approach him again and let him know he's going to have to sign a rental agreement for whatever period of renewal is left for him. Doing the dishes he dirties should be clearly included, and he should sign off on it. And if he does not stick to it, he only gets one more warning before the sheriff is called and he is evicted.
All adult human beings are expected to carry their share of the work involved in running a household.
He cannot leave his mess for others to clean up after him, and then expect anyone to respect him.
I'm sorry you're having to deal with someone like this. His behaviour is pretty childish and irresponsible.
PS: My husband, who is such a good man in so many wonderful ways, is seriously allergic to even rinsing a dish, let alone washing them or helping me keep the kitchen tidy. In spite of numerous reasonable and patient requests on my part, it just never changes. I am thinking there are just some people who have a mental block that way. The only way I can deal with it, because I cannot evict him, is to curtail how much I cook for him. That's when it usually hits home for him. (But he forgets the lesson almost as soon as he's learned it!)
Like I said, I think some people just have blinders on, regarding certain situations. You might never get through to him.
But I wish you success anyway.