OK I've got some photos and some answers...
This is a look at the parasol handle without a flash. It is a little more true to the color. The actual color is something between the white you see with the flash, and the gold you see without it.
Here's a shot with the flash showing the criss-crossing grain on the ivory. This was about the clearest shot I could get of it.
This is the end of the parasol handle. Its just part of the carving. I looked with my jewler's loupe and it is definitely not a plug, so it's not narwhal tusk.
The rod looks to be a later repair and not part of the original parasol. The break goes all the way through and the rod is holding the two parts together. You can see the glue that oozed out in this photo. Looks like a pretty amateurish repair, but it does hold it together. For now, a bow will hide it easily, and I'll look into having it redone properly.
No doubt the indomitable Mrs. Weatherby clubbed a liverpool pickpocket on the head with it as he was trying to make off with her pocketbook, whilst giving a shrill cry of "Now you've gone and done it you bloody hooligan, you've ruined my best parasol!!!"