I have a non-commercial herbicide licence, and above all, READ AND FOLLOW THE LABEL. If it doesn't have English Ivy on the label don't spray it.
I don't have any labels in front of me, but my first guess would be to trim the bulk of the ivy and then spray glyphosate (Roundup) with a little surfactant (soap) or MSO (methylated seed oil/soy oil) to make it stick and soak in to the leaves better. Manually removing as much as possible will make it so you have to use less chemical (always a good idea). Herbicides like glyphosate which rely upon leaf contact to work need actively growing vegetation to work on. If you mow it, let it start to come back before spraying. Tender new leaves are the most susceptible to contact herbicides. Also, glyphosate bonds instantly to soil, rendering it inert. If you spray it on anything but actively growing leaves, your dumping your money on the ground...
As stated earlier, many of the herbicides that we used to use have been taken away because of mis-use. Don't be part of the problem. Use the proper chemical, and apply only at the rates recommended on the label.