I agree about the landscape fabric, I think we've all found that out the hard way! If you keep decapitating the plant eventually it will die because it can't make food and will eventually starve. But this may take too long, another option would be what's called a "cut stump" treatment, where you cut off the main trunk and immediately, don't wait, apply a herbicide like glyphosate (e.g. RoundUp) or a brush killer to the cut surface. The herbicide will then translocate to the roots. There is some information in the "chemical" paragraph here on the proportions of certain herbicides to use for cut stump treatment of English ivy.
http://www.dcnr.state.pa.us/cs...
I personally don't like vinegar as a herbicide. The ones used for that are very strong acetic acid and can cause eye damage, even blindness if splashed in the eye. They also typically only kill the top growth unless the plant is very small.
