It should have a peak at 137KHz. I think your parts are not accurate enough or there are issues with your test setup.
But you can easily tweak the circuit to 137K. Just reduce the 0.16u cap until you have a peak at 137KHz. The easiest way is probably to put larger values in series.
You don't need fancy hardware for the peak detector. Just make a detector from a 1N4148 diode, a 0,1u cap and a 10K resistor across the cap. Just attach a voltmeter across the resistor.
Then do the same with the 0.005u.
Make sure you are using stable caps. Not sure what type of capacitor that blue .15uF one is, but if it is a multilayer ceramic cap, it is probably rubbish. You would probably be better of with a polyester cap. Again with the .005u, a COG/NPO ceramic would be fine for stability, but I suspect it is not from the size. You could just go for a polyester cap there too.
Now the cores you are using for the inductors vary by over 1%/C, so a 10 degree C temperature change would change the centre frequency by over 14KHz. This is fairly low frequency so you might have better stability using something like a gapped E core. With a gapped core, the stability of the inductance is set by the stability of the physical gap between the cores. The core material has little effect on stability.