The spring is too tiny and I don't have three hands. I do have at least three thumbs, probably more, but that only makes it worse.
I ended up just dropping it in the open end. It sits on the ridge that is the switch contact. Then just kinda jam it in with your thumb. It'll go in crooked and the coils will sort of cross each other. Inner turns of the spring will likely get pushed further into the tube than the outer turns, and then they get stuck. Use a battery to push the mess down most of the way, then use your hook to pull it back out to uncross the coils. It might take a couple tries.
Once you have the coils untwisted, you still need to push it all the way to the bottom. A battery actually doesn't work very well here, because it pushes on the center of the spring. You want something to push the outer edge to the bottom, and leave the center alone. A hollow tube would probably work very well. I used a paper clip, and kept peering inside to make sure I was always pushing on the edge that was closest to the top.
Everything works fine now.
edit: Oh, and another tip - When the alkaline leaks and gets stuck, don't untwist the head and then bang the tube on the table trying to get the battery to come out. It won't, and you'll just end up marring the finish on the end of the tube. You really need to drill a hole and use a screw to pull the battery out.