I think that after you chop it up you would still want to spray it with brush killer (something for woody plants, not roundup).

Goats might work. You can probably rent a brush mower instead of buying one.

Unless you use an herbicide that kills the plant including the roots, the roots will put up new growth and thank you for the mulch.

English Ivy has a waxy coating that makes herbicides less effective. It may take several applications to kill it.

I agree, you must spray it few times, run the mower over the dead ivy to scalp it, then mulch. If you want to plant over the area, beware the tangle of tough woody roots. Very daunting to remove by hand (Boy, do I know!). Maybe a BIG horse power rototiller might break them up, but you don't want to damage tree roots. If you are not in a hurry, it might be worth waiting till the roots rot a bit, and you can catch any stragglers. Good luck!

Use a good surfactant and do not use pre-mixed products if you go the chemical route. The surfactant will make it much more effective and you want to mix to a strenght for the weed you are using.
Whatever you use (if you do) please read the label, follow the directions and call the toll free number with any questions.