Bindweed, wild Morning Glory, can be difficult to control because it spreads by both seeds, moved around by birds, and roots. There are two ways to control this. Keep the above ground part of the plant cut back so any leaves cannot supply the roots with the nutrients needed to continue growth and eventually the roots will die, or dig out the roots.

OH YUK! BINDWEED! We have the same problem. The first year we moved here we thought it was just a really neat sprawling groundcover with pretty pink and white flowers. Then I saw how it climbed and choked EVERYTHING! Learned those seeds can be viable in the ground up to 70 years. Been 4 years now and it is the strongest weed on the property regardless of the organic measures I've taken so far to get rid of it.(same ones you tried). While weeding in the vegetable garden a few weeks ago, I dug as deep as 2&1/2 feet to find the mother roots. ...No luck. =(

I'll be with you in spirit during pull process. ...lol

~KraZ

At my first house, I had luck getting rid of it with boiling water, but it was over a period of a few years, and even then we still had to stay on top of it each spring. I was super vigilant about pouring hot water over all the leaves and then leaving the dead leaves on the plant. When new shoots came up I poured water on them as quick as I could. My layperson theory is that pulling the leaves/stems off just encourages the roots to send up new shoots, but leaving the dead ones on means the roots don't get nutrients and they eventually die. I have no idea if that's what really happened but it worked for me -- not a one time fix, though.

Another "weed" that is similar to bindweed is Nightshade. One bed at church has now been free of any Nightshade for 7 years because a member came over 8 years ago and dug out the roots, every root, he could find. Others pull the plants and then complain because it grows back, because they do not remove all of the roots.
Herbicides are not all that effective because both of these plants seem to be able to keep the herbicide from reaching the roots.

Recently, flaming has been studied and positive results (90% dry matter reduction) were reported.

Here is a link that might be useful: Dec 2010 Scientific Study U. of Nebraska

Here is a link that might be useful: Scientific Study U. of Nebraska

Flamming may be an alternative, if one does not mind using a non renewable resource to kill the top growth of a plant. The major problem with flamming is it does need to be repeated many times to get the results that digging out the roots once does.