A photograph would be very helpful in order to determine what the black stuff is. I'd also ask which variety of Jasmine it is, and whether its growing beneath or very near to conifers.

Could be sooty mould that is the result of honeydew deposited by aphids. Try repeat spraying with a weak solution of washing-up liquid to clear the sooty deposit and then perhaps with a systemic incecticide if there is any future aphid infestation in the spring.

Spray UNDER the leaves -you may spot some small brown lumps. These are scale insect, related to aphids.