Russian Vine - can it be killed off?

I have a Russian Vine that got completely out of control. After many hours of work I finally got rid of it and severed it at the root. I thought....

However - it's back. It appears to have re-rooted in numerous places in and around an inaccessible wall behind the garden shed and is threatening to devour everything in its path. It is even sprouting into the shed itself.

So, can anyone advise of any way of getting rid of this menace completely? As I can't get to it properly I presume I need some chemical assistance. I read on another thread that Roundup on the leaves may do it. Can anyone confirm this or offer an alternative?

Thanks in advance.