First, I want to be sure it is a Dutchman's pipe vine (there are photos on that link) and not a trumpet vine/trumpet creeper. I'm asking because I have never heard of someone battling Dutchman's pipe and it's good for me to know which vines are very aggressive. 

That said, I think you are doing the right thing. It's key that you keep removing any growth from the plant as soon as you see it. You want to starve the plant for sunlight. So pull or cut off or mow any new growth as frequently as possible.

As for the herbicide, I think that is helping. However, I googled Ortho Ground Kill and I did not see a product with that name. When you buy and herbicide, you are looking for one with the Active Ingredient of glyphosate or imazapyr. One of those ingredients should be most effective in killing this vine. Applying at this time of the year and into the fall, cutting and paining it on the stumps, should be the most effective way to kill the vine. Unfortunately, it will take time. You are trying to exhaust its reserves of energy underground and that may take years. It's a case of controlling it and constant maintenance.

If the vine in question is truly a Dutchman's pipe, you can take some comfort in that it as host for the pipe vine swallowtail butterfly. They are a beautiful butterfly and your plant is what the caterpillars eat before they turn into butterflies. So it's not all bad!

I'm sorry that I cannot offer you a better solution. 

I am sorry. The product is Ortho Ground Clear. Not Ground Kill. I typed it wrong. If I let it grow, you would see bent pipe shaped flowers hidden under the leaves. I took pictures to show my local Extension Service and other local nurseries, but I eventually deleted them. Thank you anyway.

If you purchased Ortho Groundclear recently, it may not have glyphosate in the ingredients. Therefore it might not be as effective at killing this vine. Review the active ingredients on the front of the bottle. If it does not have glyphosate, I would recommend purchasing a different product that does have glyphosate or imazapyr in the ingredients. Applying herbicide in late summer and early fall, when the vine is storing energy in its roots, is the most effective method. 

I'm sorry that there isn't a once-and-done solution. Constant vigilance is the only way to keep it under control.

Groundclear does have imazapyr. But, I am learning there is no once-and-done solution except vigilance. I have searched quite a bit about it and I seem to be the only person on the planet with this problem! lol Thanks for your help.