Why isn't my LED slowly dimming and eventually turning off?
When the capacitor is powering the LED, the circuit is an RC circuit with an exponentially decaying voltage. The LED is only going to stay lit for as long as the voltage remains above the forward drop voltage. Since capacitor values are not very precise, and the LED doesn't produce much light at low currents, we can estimate the amount of time the capacitor can power the LED as the time constant of the RC circuit. It's not the exact time, but it will give you a ballpark estimate of the decay time for the fade out. The time constant for an RC circuit is the product of capacitance and resistance:

$$
\tau=RC=(220\Omega)(22 \mu \text{F}) \approx 4\text{ms}
$$

The time to fade is only about 4 milliseconds! That's why you aren't seeing the LED fade off, because it happens too quickly. There are two things you can do. The first thing you might try is increasing the size of the resistor. This will make the time constant longer, but it will also limit the current, making the LED dimmer. The other thing you might try is using a larger capacitor. This will increase the time constant without effecting the brightness of the LED. You can figure out the value for yourself, but I'm guessing you want a fade out time of around 1s, which means you need a capacitor roughly 1000 times larger than what you have currently.