You could try smothering it, either with layers of newpaper, or with landscaping fabric. Weight it down at the corners, engage you patience. I am not sure how long it might take - maybe a couple of months??

Good Luck.

Bless you and I wish you luck. I had ivy in Nevada. I will list what didn't work. These attempts were after the first surgery of sawing down the 8-10 ft vines in a 1ft by 5 ft (fortunately well bordered) strip

Three weekly drenching of Total Vegetation killer followed by:
Layers of rock on top of newspaper
Black fabric- both newspapers and fabric were considered LUNCH!
Soaking the leaves in motor oil (we were getting a bit frustrated at this point-- i mean buckets of oil with STALKS sitting in them for a week!)
Digging was mixed in at many times, when we were fortunat enough to see the ground

Eventually we sold the house and moved to a place with seasons so I'm afraid that I never did find a good way to kill it