Answer: My preferred method is to dig out the roots. Water the area deeply a day or two before to loosen the soil. Glyphosate, the active ingredient in Round Up (and others) is a systemic herbicide used on English ivy. However, it should be sprayed on foliage when it is vigorously growing, so it is absorbed and circulated through the entire plant system, including the roots. I don't know that it is effective absorbed if sprayed only on the root system. You could read the instructions to see what it recommends. However, since you have torn down the green foliage, if you monitor the roots regularly, and immediately remove any sprouting material, the remaining roots will eventually "starve" because there is no foliage to photosynthesize.
The Ivy Removal Project in Portland, Oregon has a website devoted to this topic:
Q&A Library Searching Tips