colpitts oscillator does not work with small inductance
You might try using the resonating inductor to feed DC voltage to the transistor collector in a different arrangement of so-called &quot;Colpitts&quot; oscillator:

 Be aware that using a 100uH inductor at 10 kHz gives a low-impedance resonator. If you want decent output amplitude at the collector, a lot of collector current is required.
Don't forget to add a large capacitor from the top of the resonating inductor to the bottom of those two resonating capacitors (C4, above). Much of the resonating AC current will likely flow through this capacitor.

A larger-value resonating inductor would improve this oscillator.