Cindi and Amanda ... the good Doctor was used as rootstock for a very good reason. It is excellent at growing roots !
Think of it as the bindweed of roses. It will grow new roots at every node. No, I am not kidding.
As much as I truly hate to use any kind of weed killer, I will use it for poison oak and Dr. H as needed ... oh, and for blackberries that are ladder fuels for wild fires which is simply due diligence in my part of the country. Some people rent goats to clear their blackberries, but with mountain lions around, I'd rather not rent bait.
To kill Dr. H, you need a container that you can stick several canes of Dr. H into during the active growth period of the plant in spring or during the fall season when the plant is preparing to go dormant. Fall is the best time because the plant pulls the sugars and saps from the top growth down to the roots as it prepares to go dormant. YES ! The plant does the work of placing the toxin where it will do the work needed to kill the plant.
You want to make sure the container is properly fenced so that no other critters or children can have access to the container. It's temporary ...
For a well established root system, I suggest you do the application both in spring and in fall.
If you get intense summer heat like I do, you can even cover the plant with black plastic bags weighted down with rocks and cook it.
I am usually treating poison oak with this method because that stuff can put me in the hospital, so drastic methods are required, but it will also work with Dr. H.