Answer: I'm afraid Mondo Grass needs more sunshine than the site can provide. Here are a few suggestions for groundcovers that will grow in shade; Houttuynia cordata, Star Jasmine, Vinca (periwinkle) and Parthenocissus. Be sure to prepare the bed well prior to planting by amending the soil with lots of organic matter to help it drain quickly yet retain moisture for your shade-loving plants. Water deeply once each week, and your plants should thrive.
As for the ivy, you're certainly within your rights to remove any that escapes into your yard, but I wouldn't recommend using an herbicide to kill it. Anything you spray on the ivy could affect the parent plant and you wouldn't want to be responsible for killing something on your neighbor's side of the fence. Cut away the ivy and then install a barrier on the property line to keep the ivy from invading your yard. You can dig a narrow trench and use galvanized panels (like those used for patio roofs). Dig down at least 24" and set the panel on its side so it becomes a solid barrier and ivy roots or stems can't get through.
Best wishes with your garden!
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