One of the neat things about this is that you can test it with fifty feet of wire and an old radio. Place the loop of wire around the room and drive it with the speaker output of the radio (or an old tape recorder or something).
Start with the amp circuit, don't bother with the resonant cap yet. Hook your detector coil across the differential inputs to the op-amp circuit, connect earphones through a cap to the output. With high gain (use a pot on the feedback resistor of the opamp so you can adjust it) when you bring the detector coil close to the loop, you can hear the output of the radio. This way you can test the first stage of the circuit.
You can also listen to the 10khz signal this way.
With the resonant cap, you will improve the detection range and be able to back off on the gain, which will reduce noise and interference. It will also be more selective as to what frequency it detects - the radio will sound tinny and lack bass.
At this point you can rectify the output of the amp with a charge pump (two diodes). Drive the pump through a capacitor or you will get the offset of the amp. Send the output of this to a cap and a bleeder resistor (use a pot for the bleeder so you can adjust it). Now you can measure the voltage with a meter, see if it acts the way you want before you write code for the A/D.