I've been putting gauze pads on my toe and taping it down to keep the broken nail from moving around or getting caught on pants legs or my socks. It was working quite well. I'd cut the nail twice so far and resumed the bandaging process. Yesterday morning, however, I cut the nail for the third time and when I was done, I noticed that the broken part was no-longer hanging on by the tiny amount it had been before.
I simply lifted the end of the nail off and threw it away. It didn't really hurt, I think the nail bed has had time to toughen up. To my surprise, the nail has grown back even more than I'd thought. There's only a little of the nail bed still exposed. I'd say there's still 1/2" to go, but there's a protective, thin layer of nail over most of that area.
I'm still doing the bandage thing for awhile, but I should be able to stop even that soon. I think the new nail is growing in healthier than the old one. Wish me luck!
Everyone else can.