Why does my Current Regulative Diode not regulate current well?
It's a single JFET, and probably a resistor, packaged as a diode. This won't be very accurate. The "voltage reference" used is the JFET's threshold which depends on temperature, which depends on voltage and dissipation. This, plus other JFET properties, mean it will vary quite a lot. It's still decent if you want a somewhat regulated current, with one part, and it is much better than a resistor if the voltage will vary a lot. It is not adequate for your application.

If you need something a lot more accurate and stable, try this:



TL431 is cheap, readily available, low noise, accurate and stable with temperature.  It will adjust the current drawn from pin "A" to keep pin "REF" at 2.5V above pin "K", via feedback through the internal error amp. This adjusts Q1's base voltage to control LED current. 

I wired the BJT this way to make sure the base current (which depends on temperature and BJT) is not included in the regulated current.

For a 10mA current in the LED, set R2 = 250 ohms. Adjust accordingly.

This will cost maybe 20c and result in excellent stability. Main source of LED brightness variations should be air currents changing the LED temperature...