Another way to go is with one 2N7000 or 7002 MOSFET and nothing else other than the R and C. No schematic capability right now, but here is the circuit:
R (2.2 meg) and C (1000 uF) in parallel from transistor gate to GND.
Switch from gate to Vcc.
Source to GND
Drain to LED to resistor to Vcc.
When you press the switch, the cap charges up to 9 V almost instantly. After the switch is released, the resistor slowly discharges it. This is in addition to the capacitor self-discharge due to leakage current, which is why the resistor value is larger than expected. When the cap voltage gets down to around 2 V, the transistor will begin to turn off. The LED will turn off slowly over about a minute.
ak