I don’t know if you could break up the roots of ivy with a rotary hoe. It depends on the size of machine that you hire. If it has enough horsepower and strong enough blades, it might do the job. Talk to a hire company. But what then?
Every cut root and stem then has the potential to become a new plant. This is because stems of English Ivy that grow at ground level are referred to as “rhizomes”. This means that everywhere that a leaf would normally grow on the stem (the leaf node) has the capacity to produce a leaf or roots or both. So having chopped the roots and underground stems (rhizomes) into little pieces, instead of having a mesh of roots, you now have a mass of new plants that are growing roots and shoots. If commitments stop you from getting back into the garden for a few weeks after using the rotary hoe, these thousands of cuttings that you kindly planted back into light and fluffy cultivated soil could then be thousands of plants.
You could use the cultivator and then sift every last bit of soil to remove all the roots and stems. With 1 or 2 old fly screens you can lean them at 45 degrees against a wall and throw shovel loads soil against them to sift out the Ivy. Another option is to get a couple of friends to hold each end of the fly screen and shake the soil through and put the Ivy in a bin bag. You certainly can’t bury it under the soil. It won’t die but will come back like The Day of The Triffids.
I could make chemical treatment recommendations but it requires that the Ivy is actively growing. While chemical treatment isn’t my preference, it has its place in large scale weed infestations.
If you have garden waste bins in your municipality, no one will thank you for putting thousands of English Ivy cuttings in it. Usually this material is composted and sold back to gardeners. English Ivy is more likely to grow in the compost than be killed by it and it will then get transported to other people’s gardens. Landfill or drying and burning is preferable.
Finally, if you choose a labour intensive, chemical free method, I would strongly advise you to also invest in your friendships and buy a slab of beer and a few kilograms/pounds of BBQ sausages. Get a group of friends around and knock the job over in a day. Make sure that not one beer nor sausage is available until the end of the day or when you are satisfied that you can finish the job easily the next day. Where there is free booze and food involved, you need to motivate the troops to get the job done and done well.
We can often find it difficult to ask for help but I firmly believe that they are very happy to and more likely to ask for help as a result. Then there is a chance to become a group of friends who help each other out with the practical stuff. Isn’t that how life should be?