I spent some time yesterday trying to find something to do with these - most of the recipes I found were from the USA where the fruit seems to be used as an ingredient in baking (cakes, sweet breads, desserts etc). I've printed a few off and will try them over the next few weeks as the fruit ripen.
There was one recipe which was called Persimmon "Butter", in the photo the consistency looked like lemon curd (but a rather unappetising brown colour). I've not had much success with jelly's in the past (not the jelly and ice-cream type, but the "conserve" type) - mine tend not to set!
I shall keep searching and experimenting, and if I find anything worth passing on then will do so.