Not what I wanted to hear, but thanks!

Any thoughts on what to do while we wait for the ivy to die/actively try to kill it? Thick mulch? Cover with a tarp? Scorched earth? (Don't worry, that was a joke... kind of...) I think if we leave it and just keep pulling roots, it will get really eroded. Again, it's a pretty patch of dirt.

*Pretty BIG patch of dirt - whoops!

In my experience covering Ivy with a mulch simply provides it with the environment it likes best, cool, shady, moist, and it grows even more prolifically then with no mulch. The only thing I have found that will, eventually, eliminate Ivy is to keep it cut so new leaves cannot provide any nutrients to the roots. When the roots are, eventually, starved out they Ivy will no longer be.
Since there is Ivy growing in your neighbors yard you will need some kind of barrier to prevent new roots from invading from that source and it needs to be high enough to keep stems from growing over it because those can root as well.

If indeed the area is bare dirt, your "scorched earth" idea might be a way to accomplish kimmsr's foliage removal using a weed burning torch (they do make 'em)! Incinerate any emerging foliage weekly...also melts snow, peels paint.
hortster

Here is a link that might be useful: Torch

Oh man does the torch sound like fun! :-) Thanks again for all the knowledge and suggestions. We're still such beginners!

25x25 is admittedly a fair sized area, but have you tried digging? English ivy, as opposed to Boston ivy, has a fairly shallow root system. I don't know that I would try planting grass there the first year its been removed, but I could see the possibility of establishing a new tree while you wait until later to finish landscaping the area - if you had some time a few afternoons or evenings to do some digging (notice I didn't say do it all in one weekend :)) You might find a digging fork easier than a spade.

Then carefully watch for any new shoots from missed roots or germinating seeds, pull sooner rather than later. I pull ivy seedlings every year, nearest ivy is a block away from me.

Here is a link that might be useful: English Ivy WA State

If you have a relatively thin patch than just pull it out.

If you have a dense patch than it is harder to remove. A dense patch will have started to form a web of vines that makes it harder to remove. Once it forms a tangled web of vines it is going to be harder to pull and more prone to snapping as you pull it because the interlocking vines form a tanlge and secure each other to the ground. Pull on a vine in this web and it will snap rather than uproot, leaving healthy fragments intact. However, with the exception thick roots (of perhaps .3" diameter or more), English Ivy is not that resilient to abuse and a dense web that is hard to remove by pulling can first be knocked weakened by weedwacking until all the leaves have been removed from the web of vines. Let it rest and weedwack again 1-2 weeks later. Wait another week or two and start pulling the vines the day after a heavy rain as vines are less likely to snap when pulled from soft wet soil. As you pull them you will find they tend to snap where passing under an intersecting vine. Dig your finger a half inch or more under the surface at the location of the break to find the intersecting vine, and uproot that one as well. This method will help you find vines just under the surface and break up the 'web', and makes for more efficient uprooting.

I've removed these vines from several large areas.