Firstly, I doubt your circuit would produce \$V_{BE}=-5.8V\$. I suspect your transistor is damaged, isn't what you think it is, or is wrongly connected.
With such a small resistance at the base, most of the collector current will flow out of the base, instead of through the LED. You'll also need to be more explicit about how you set base potential. Both problems are fixed here:

simulate this circuit – Schematic created using CircuitLab
R1 and R2 set base potential at the mid-point between 0V and the battery voltage, for about +7V. Those resistances have been chosen for a base current which is very small compared to collector current. This places the emitter slightly above that at about 8V. Emitter and LED current are then:
$$ I_{LED} \approx \frac{V_E}{R_3} = \frac{13.8V-8V}{800\Omega} = 7mA $$
LED current will vary slightly with battery voltage, since emitter potential will vary with the base. To mitigate this, the base can be held stable with a zener diode:

simulate this circuit
Unlike the resistor divider, whose voltage varies with the battery, the 5.1V zener diode here keeps the voltage across R3 stable at 4.4V or so, over a wide range of battery voltages. This keeps LED current constant:
$$ I_{LED} \approx \frac{5.1V - V_{BE}}{R_3} \approx \frac{4.4V}{560\Omega} $$
That seems a lot of work when the following would do exactly the same thing:

simulate this circuit