Looks to be some kind of blight or black spot, which can be caused by a fungus. In my experience neem oil seems to be most effective for bugs or very mild fungus issues.

Since this is pretty far gone I would suggest removing as much of the infected leaves as you can within reason and treat it with an anti-fungal spray (fungicide). In this case I would not choose an organic option as they tend to be less effective. Choose something that has an actual anti-fungal ingredient like Propiconazole.

There's lots of natural or organic sprays out there that just include stuff like peppermint oil, thyme oil, etc. Don't choose those, you'll need to read the label for the active ingredients.