chemenger has the right idea I think. Get the cleaner to stop doing the washing up, make this public knowledge at the next staff meeting, also follow this up with written memos.
First memo- Dear colleagues/ attention please colleague,
The cleaners duties no longer extend to washing up. S/he is going to concentrate on cleaning the office space, including the kitchen.
So, WE need to wash up. Daily. The unwashed cups/ plates/ cutlery (that we are currently leaving for the cleaner to tackle) are building up in a way that is unsanitary and antisocial. It is bad for the office, the people who work there (ie us) and our clients. Please can each person ensure that they wash and put away all the cups/ plates/ cutlery that they use.
Yours, Threemoredays etc
Be prepared for this NOT TO WORK for a while. Everyone will ignore it. Ingrained habits take a long time to change.
So write a second memo- Dear colleauges, Re the washing up. We still have unwashed cups/ plates/ festering food all over the place. It's a problem that needs to be tackled. Please make sure that you wash up after yourself. Thanks. etc.
Repeat. Send individual emails out, to everyone, about the issue.
Then (after this and subsequent emails, which still won't have worked), send a memo and announce at staff meetings that all dirty crockery will be thrown away on a certain day. And then follow through, and actually throw away all dirty crockery. Be prepared for a lot of anger as people have things thrown out that are wanted/ special. A lot of people will lose cups that are theirs, but used by someone else. Everyone will hate you (except for the possible silent majority who are as exasperated as you are about the situation).
At this point, some people might see the light and come over to your way of thinking. But many probably won't.
Repeat the whole exercise again and again at regular intervals.
It probably won't work in the long run. But it'll be an interesting challenge!