An even simpler timer, for this type of application, is a large capacitor in parallel with a resistor. I have lots of capacitors lying about, so I just use trial-and-error, but if you don't have that kind of stock at hand, you may need to figure out the specs to see what size cap and resistor you need.
For instance: If I pair a 2200uF with, say, a 47Kohm resistor in parallel, and put that in series with an instrument LED (which I assume has its own resistor in series) I'd expect it to light for several seconds before the capacitor charges and stops the current flow. The 47K resistor in this example was usually high enough to limit the voltage at the LED so it won't light.
When the voltage source disconnects, the capacitor discharges through the 47K resistor and it's ready to go again.
If you know the current draw of your LED, then the values you need for the capacitor & resistor can be calculated.