- Any smart tips for how to eliminate it?
- Can I compost any part of it, or does composting even, say, the leaves risk its propagating itself in the compost pile?
- I know I need to remove all roots, but what about the little clumps of white root hairs that extend off? It will take significantly longer to sift through the soil and get all these, but I want to do this only once, so I'm willing if it's necessary.

Thanks!

I wouldn't put it in an open pile without killing it first, which I would definitely do. You're going to earn that organic matter, it might as well be put to work. You could kill it in a plastic bin or bucket like a trash can, a trash bag, or even laying exposed on concrete or sheet metal to bake in the sun. After a point, it will definitely be dead, then you could put it in the compost to finish decomposing.

It's not flowering and producing seeds is it?

About the tiny roots, if you are able to smother the area with cardboard or newspaper after you are finished digging and leave that alone next summer, you should be able to be sure that any ivy that attempted to keep growing was smothered under that layer. Cover with mulch or leaves so it's not an eyesore and stays in place. Maybe set some potted plants there next summer. Then you are free to garden the area at will after that. If that's impractical, it should be fairly easy, assuming you've done a thorough job of removing the bulk of the roots, to spot-dig little pieces you might have missed that throw up new sprouts. They might be so loose you don't even need to dig, just pull. You'd want to check it often next spring.