How to make a current source that supply constant current independent of a voltage power supply?
My suggestion would be an Nexperia NCR401T as long as you can accept a maximum Vf of your LED being limited to 5 (minimum input)-1.4 (dropout) = 3.6 V.
A three terminal device but no external components needed.

Datasheet
Another suggestion would be a single JFET, which you need to pick (Perhaps hand-pick if your tolerance is too narrow) for your 10 mA.

Going up to two components, you get a little bit more refined solution if you add a resistor so you can adjust your JFET solution to get your desired current.