So I assume you had the top nut off? I would have that off for sure when soaking it. I don't think the shaft goes past the middle and if it does it can't by to much. I know the shaft will not exit from the top. So to get it in, I would think at least the top outer section would need to be unscrewed from the middle. The trick is to get it off without messing up the nickel plating - how to do that, I don't have any ideas.
I've always wanted a small strap wrench for things like this (and say for the smoke stack too).