Only digging them out. Weed killers run off the shiny leaves. Best you can do is just keep pulling out the long stems. If anybody else has a sure remedy I will be grateful to hear it!

Not sure if it will work for you but I once had a Stags Horn tree ( Rhus typhina) in the garden which when I pruned it back hard started to throw out shoots which eventually were coming up all over the garden in the lawn & the other side of our path. I chopped down the tree completely, but the damage remained in the lawn.
After many tries at pulling them up which was causing even more damage to the lawn, then, after reading as much as I could used SBK with a little engine oil mixed in a bowl and with a small paint brush started the lengthy job of painting (very carefully) the leaves that were in the lawn and those that were coming up the other side of the path. It took two complete years for them to die back and not return. I also poured a mixture of SKB & engine oil in the middle of the stump after so that it didn't grow again. If you can find the main growth source or at least one of them it may help to quicken the procedure
If you do decide to try it don't pick windy days.

I then spent time on my lawn and you would never know that the problem had ever happened, it took a lot of patience but was worth it. Good luck. mace

Trouble with ivy in hedges is you usually get a lot of thin stems which would take a lifetime to sbk. If you go for painting the leaves you'll need to add some detergent to the mix because they have a waxy surface and liquids just run off. Does it matter a lot? It will make some evergreen cover in winter and the birds will enjoy it. Sorting it out from hawthorn is a bit of a nightmare prospect isn't it? Natural hawthorn hedges often have a mix of ivy and wild rose.

I would try 'Roundup' (Glyphosate) mixed with wallpaper paste and applied above and below the leaves with the fingers of a cloth glove (wearing a disposable glove underneath). If you apply on a warm day it will adhere to the ivy leaves. You might have to repeat several times but it's worth a try.

Good idea Jimmytheone. Glad its not me diving under the hawthorn though!