Why doesn't this Colpitts oscillator work?
I can say with absolute certainty that you have ruined the transistor biasing by not having a capacitor fitted as shown below: -



Without the capacitor shown above, the DC voltage at the base will be about 3.7 mV and this stops it working dead.

And, I can see that it is intended to operate at 10 kHz but, even with the capacitor in place, it might still not work if you have used crappy electrolytic capacitors. Other than that, it's really hard to make this very simple common-collector Colpitts oscillator not work.

This answer provided to you on your previous question shows the role of the capacitor I have added to your circuit: -



Notice the phrase "DC block"?