Ideally, the neighbour would take the responsibility for their land and fix the retaining wall. I suspect they won't do anything until the whole thing collapses.
You might be able to go halves fixing it up. Although I know these situations can be messy if the neighbour isn't willing to do anything. In NZ, that retaining wall would be entirely the responsibility of the neighbour because it's his land. The fence would be dual responsibility and dual ownership.
If you haven't already done so, I'd first approach the neighbour.
See if you can get the water issue dealt with in a way that reduces the erosion. Maybe you both can work together to fix it. It's in both of your interest to do so. Get your wife to bake some muffins to help sweeten the deal? Then see if he'd be willing to also strengthen, fix or replace the wall. Again you may have to work together on that one.
Another option you could take is to think of what you'd do if this was a bank next to your property and there was no neighbour there at all.
If that was in your mind then you'd build a new retaining wall in front of the old one is a perfectly valid option.
Going by your pictures it would appear that there isn't much of a footing on that wall and that it could collapse at any time.
I'd build the new wall just off the old against the old and add a bit of a slope against the face of it instead of being vertical, so that the base is wider than the top.
Allow about a foot between the old and the new. Partially fill with p-stones in the void between the walls. Then add drainage pipe - the type with holes in it designed for draining the ground. Then cover that with more p-stones and top off with soil. Put some plants on it and make it look nicer.
What that'll do is it will stabilise the wall. You'll loose a bit of land, but at least it won't fall away.