Okay... I would target a maximum LED forward current of 25mA. Easier on the LEDs, and the 5mA difference in candlepower isn't significant IMO. It also gives us a margin to work with.
With a 12V 500ma adapter, and 25mA per LED, that max number of LEDs to connect is 20. There are a few ways to power more LEDs than 20. Get a bigger adapter, use more than one adapter, or connect 2 LEDs in series with each resistor. This third choice will affect operation though, because the LEDs will turn off at a higher voltage (~6.6V instead of 3.3). And if one LED goes bad, the other goes out too, like xmas light strings.
If you want the pot to totally dim off the LEDs, you don't need a resistor where the 680 ohm is located in the linked circuit. 500k is rather large for this circuit, and I don't really know what effect that may have on the regulator. It might end up giving a lot of "play" in the pot's action on the output.
The 12V adapter and 470uF capacitor will provide about 17 volts to the input of the regulator. Figure the maximum regulated output will then be 15 volts. Using the formula from before, with 1 LED and resistor in series, the resistor needs to be about 470 ohms. For 2 LEDs and a resistor in series, the resistor needs to be about 360 ohms.
The regulators will need to dissipate 5W or more. A small heatsink for each would certainly help. They don't need to be store-bought. Large washers and/or "wings" cut from a beer can would probably work fine.