You don't have component designations on your schematic, but I can see some things wrong with it.
Transistor on bottom left is PNP, yet it's emitter is connected to 0v. It will not conduct this way.
Transistor at far right has it's base and emitter connected together. It will not conduct this way.
LED is in series with the emitter of the transistor at far right. This will provide negative feedback to the transistor, which will tend to counteract the effect you want.
I get the impression that you want to press the button once, and the effect continues after releasing the button. The circuit you show won't achieve that because there is nothing to keep the transistor at far right conducting after you release the button.
There is no current limiting resistor for the LED. This not only protects the LED but also in this circuit would be the means to control the time over which the capacitor discharges - the time over which the glow decays.
I think your friend here is the humble 555 timer. There are hundreds, if not thousands, of example circuits you can find online using this chip. I suggest you investigate 2 possible approaches:
1) set up the 555 as a monostable. Set the pulse width to be somewhat longer than you want the glow to last. (Maybe twice as long? I'm not sure) Connect a resistor (lets guess 47k) to the 555's "discharge" pin, and connect the base of a transistor, lets say 2n2222, to the other end of the resistor. Connect the emitter of the transistor to 0v, and connect the LED in series with it's limiting resistor to the collector of the transistor. Connect V+ to power the LED. I think this will do what you want, but it is only in my head so it might not work or might need changing somewhat.
2) set up the 555 as a bistable. Normally you have a "set" and "reset" switch. Connect the output so it charges a capacitor via your LED and it's current limiting resistor. Instead of having the "set" switch, connect the capacitor to the 555's trigger input. This way, as the capacitor charges, the LED will light fully and then dim as you require, and when the voltage on the capacitor reaches 2/3 of the supply voltage (one hopes the LED is dim enough by then) the timer will be set again, halting the process. So the circuit is triggered by pressing the "reset" switch. Unfortunately with this arrangement you can't use the "discharge" pin to discharge the capacitor as it might seem because it follows the output. You could add a transistor to do the job though. Again this is off the top of my head and I only think it will work, and really needs designing properly.