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The obvious choice will be something like the LM317, but if you don't have one at hand, you can make a very simple current source like the circuit shown below. If you make R1 or R3 variable, you could adjust it between 10 mA and 20 mA.
The switch is just for illustration of the minimum and maximum range.
Here's the LTspice simulation of a two transistor circuit that should do what you want.
I didn't have a model for your µV LED so I used a 6.2V zener diode (D1) in series with a junction diode (D2) to simulate the LED's 7V drop.
Adjusting the pot over its limits gives a constant current range of a a little less than 5ma (R1 in series with the maximum pot resistance) to a little over 21mA (R1 only with pot at zero resistance). - - - Updated - - -
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