Simply pick off any brown leaves and trim when they grow out of bounds. Ivies can be easily propagated by rooting tip cuttings. This is typical of a plant with spider mites. The insects are so small you may not notice them, but they leave the leaf surfaces rough and in severe cases, you will see a fine webbing. Regular misting or washing of the foliage will discourage spider mites.
Sooty mold tends to grow on the sticky spots of honeydew that is exuded by aphids.
Try wiping some off and see if the tissue beneath the soot is affected. Assuming it isn't, look for aphids and treat that problem. This is another insect problem called scale. Wipe them off with a cotton swab dipped in rubbing alcohol and use a systemic insecticide. Occasionally, ivy will get a few aphids.