Powering WS2812 LEDs (separate or from board?)

Then it's junk. What next?

I think that you should probably buy what you need: a good 5V supply. Do what's right, there is no magic bullet.

You could make a 5V supply from that 12V module; a regulator, a couple of capacitors and probably a heat-sink. Here again - more stuff to buy.
As you're asking these questions, that may be outside your skill set.