Geoff Lawton has a short video about sheet mulching (probably a lot more than one, but I remember this one):
Seems like you'd put down the soil and any organic material you have, then overlapping cardboard that's been soaked in water, followed by the mulch. I could see a heavy downpour washing the material off the cardboard, so if you can put a little edging of any sort on the bottom of the slope, and add additional material on the lower side to level it a little, that should help.