Here's a simple "napkin design" off the top of my head. I can't imagine it getting any simpler. As Einstein said, "Everything should be as simple as possible, but no simpler." The critical part here is Q1, a very low threshold N-channel MOSFET. Parts like the TNO702, TNO604, and ZVNL110 are great choices because of their typical +1 V gate threshold spec. Spec "Vth" is the gate voltage required for turn-on of the channel. This sets the minimum peak audio level required at the input. With a 1 V threshold, the LED will turn on around +1.6 V peak input (roughly correspond to a +3 dBu sine wave). There may be lots of other suitable parts, but those with higher Vth will require more input level. All diodes are general purpose types like 1N914 or 1N4448. C2 must be a plastic film type for very low leakage current (electrolytics of any kind or most ceramic types are too leaky for this). R3 and C2 set the time the LED remains on after audio goes below threshold level. R4 is calculated for about 2 mA in the LED, fine for high-efficiency LEDs - and also keeps battery drain low. D1 keeps the input signal from driving the gate too high and D2, along with C2 and R3 form a peak rectifier. R1 is to minimize any distortion that might be added to the monitored line. I figured 9V batteries are everywhere guitar players and stomp boxes are, so chose that as power. Obviously, you can substitute or adapt design for other voltage but wouldn't recommend anything lower than about 5 V.