Time constant calculations: [Link: en.wikipedia.org]

The trick here is to work from the current draw of the LED strip and convert this to an equivalent resistance that will be used to draw an equivalent circuit while you calculate possibilities. You'll need to fuss with things because the human's response to the change in light value is nonlinear, as is the voltage ramp, and the LED's will not turn ON until the voltage crosses their forward drop (probably in the 1.8 - 3V range, depending on color). I would start with a charging current that is twice the rate of the LED current draw.

If the DC supply is regulated, this will mess with the timing because the regulated supply probably has its own capacitor and the supply's output will not collapse instantly. However, you can work around this by loading the supply, thereby forcing it to collapse promptly after power is removed.

If the capacitor does not discharge fast enough, place a reverse diode across the charging resistor, and a resistor in series with that diode. After the charging supply collapses, this reverse R/C diode network, along with the LED draw, will control the discharge rate.