Metal detecting using a small inductance
It will be a problem getting 15 uH to resonate below 10 kHz but you can always run it at several hundred kHz and divide the frequency down with digital dividers. I'd recommend using a colpitts oscillator and there are plenty of examples on the web: -



Feed the output to a comparator (maybe a TLV3501) and then onto logic chips that can reduce the frequency to below 10 kHz (maybe a 4040 clock divider).