Is your side the ivy side? or both sides? The ivy looks well manicured. Any ground cover will have the same issues as the ivy. How much work are you willing to put in after getting rid of the ivy? A shade garden with perennials/shrubs will take ongoing maintenance but would be beautiful.

Getting rid of ivy is difficult. Mix soap with the brush killer to help it stick to the leaves. Spray this month or beginning of next month. After the above ground ivy dies back, mulch and wait for the ivy to come up from the roots in spring and re-spray. If you don't want to use chemicals you can mow close, cover with cardboard or many layers of newspaper and a thick layer of mulch to smother the ivy. You will still have some pop up that will have to be dealt with.

After the ivy is gone it would be a great place for hosta, ferns and other shade perennials, and blooming shade loving shrubs. If you are near Atlanta the Grower's Outlet is a great place for plants that are reasonably priced.