Awesome, :ty1:
I ran Inkscape from the command line and it told me the following.
terminate called after throwing an instance of 'Gtk::IconThemeError'
Emergency save activated!
Emergency save completed. Inkscape will close now.
If you can reproduce this crash, please file a bug at www.inkscape.org
with a detailed description of the steps leading to the crash, so we can fix it.
Aborted (core dumped)
So I went in and changed my icon theme and the error went away, I can now see the "Layers" dialog. Didn't need to reinstall Inkscape or anything.
I was originally using the Numix icon theme. I tried out all of the icon themes and I have installed, and that's the only one that seems to be problematic. While I'm not really sure precisely what "Gtk::IconThemeError" means, this definitely seems to be a problem with the Numix icon theme and not with Inkscape.
Thanks again for the help! I really appreciate it.