I had the same situation when I moved onto my property about eight years ago with multiflora roses, japanese honeysuckle, tree of heaven (hell), barberry, oriental bittersweet, and more invasive, tough plants. Basically what I did was I worked on a few feet's worth of that invasive hedge at a time, like in four foot sections I guess. I would cut it about a foot from the ground and soak it with brush killer. Then I would wait for re-growth and soak again. Until I saw an end to the re-growth. Then I would move on to the next section.

Not only did that make it manageable for me to work on, but it gave any animals that were relying on those brambles a chance to find a new home. I'm slowly replacing that hedge with native shrubbery.

Oohh, tough. I know exactly what you're dealing with, being in MD myself. I think my normal methodology, if I had consecutive days to deal with it, would be to go from one end to the other. But if I had to do it in stages, I'd go for the flowering vines first so they don't get more chances to multiply. If I were going to keep the original hedge then I'd worry most about getting the vines off the hedge, and then worry about killing those vines later. I'd also pull the ailanthus quickly, if they're still small and you can get the whole root. Once established it's hard to kill.