Hi dianthus,
If you can just about reach the tips of the ivy, it may be possible for you to paint the ends of the runners with a systemic weed killer such as 'Round Up' this will enter the sapflow of the ivy killing it off right down to the roots.
Thinking about this, what I'd do is gather the runners together into managable bunches then snip off the tips and then paint on the Glyphosate and it may also help to fasten a pollythean bag over the bunches. Take great care. Tbird+