[Dan in Pasadena]

Personally, rather than try to get rid of all that ivy which will be difficult on that sloped terrain, I'd design/add pathways and perhaps a deck or two. I can't tell the severity of the slope but a mix of decomposed granite pathways or redwood platforms 7 steps might looks very nice.

I'd augment that in the landscape (since it looks predominantly shady) with shade loving plants. Hosta's, ferns, some shade loving ground covers. Even if you could get rid of all the ivy you're still going to have an extensive maintanence job on that slope with WHATEVER you replace it with. Why not let the majority of the ivy remain and serve the purpose of continuing to retain that slope while adding texture to a landscape design?