I'm unclear how they made that mirror, [...] They explain how to make small conical mirrors, but the one shown below seem to be large and of high quality.
I'd guess they bought the larger mirrors, but they're not hard to make if you've got the tools...
Make or buy a cone of plastic with a smooth polished surface.
Make a vacuum chamber with some heating wires made of wolfram (tungsten).Wolfram is extremely brittle, so handle with care.
Make some short V-shapes from 2..3mm aluminum wire and hang them over the heating wires.
Place your extremely clean cone in the vacuum chamber, equally surrounded by the heating wires (in a distance) with alu-V's.
Seal the chamber.
Heat Wires to melt the aluminum a bit (it will generate an aluminum "mist").
When the right temperature of the aluminum is reached, escape the vacuum suddenly.
Your cone is now covered with a thin mirror finish that you should newer touch or try to polish.
Practice on scrap parts until you get temperature and amount of vacuum right.
That's how mirrors for astronomy telescopes are made (only they vacuum deposit on thick glass slabs of more than 2" thickness for stability, after weeks or months of hand buffing the glass).
You can't polish aluminum to a finish that comes even close.
If you'll settle for less optical perfection, buff the cone and spray it with conductive paint in a very thin layer, Add a layer of copper electrolytically and then a layer of nickel.
If you know of a place where they put chrome or nickel on plastic parts, you may be able to get them to do it.
Don't touch those kind of mirrors either. Even if they're harder, it will impair their optical qualities.
Another option is to slaughter a cheap Chinese LED camping lamp, they have a small (5..8cm) conical mirror to make the light cover 360° horizontally.