The circuit should have worked when it was connected to your headphones jack.
The circuit uses an LM386 power amplifier for a preamp instead of using a low noise opamp. It might have too much gain so maybe C2 should be removed.
C1 looks like a polarized 0.47uF electrolytic capacitor. Its value is too high so it rolls off frequencies below 3.4Hz. It should be a 0.22uF film capacitor which will roll off at 7.3Hz.
R4 and R5 set a peak input voltage of 4.4V to turn on the 10th LED but diode D1 messes up this peak voltage requiring it to be 5V. The diode also messes up the range since the first LED should light with an input of 0.2V but instead the bottom 4 LEDs do not light at their expected very low levels (certainly not 20mV).
The diode should be an active peak detector circuit (using an opamp) shown in the datasheet for the LM3915.
A stereo to mono adapter is simply a 10k resistor from the left channel and another 10k resistor from the right channel. Join the outputs of the resistors together at the input of C1. Join the shield of the cable to the circuit's ground.