Resistor values calculation: how to arrive at these values? (Make: Electronics example 10)
The computation has to be done as follow:

Referring to your diagram, the current that light up the LED follow this path:
from +12V  -> R1 -> Q -> R3 -> D to -12V. (you can neglect the current from the base of Q)

The total supply voltage is 24V (from -12V to 12V)

The value of the resistor is set to make sure that in worst case conditions, the current is kept below the 20mA max of the diode.
This worst case condition is when the transistor is in saturation (when it has the highest conduction between the collector and the emitter). In this case these is a voltage of Vce_sat between the collector and the emitter. It could be as low as 400mV according to a datasheet found on the web. 

Thus the voltage across R1, Q, R3, D in series should be 24V when there is a current of 20mA.

$$ 24V=R1*I+Vce_{sat} +R3*I + V_{diode} $$
$$ 24V=(R1 + R3 )*I+Vce_{sat} + V_{diode} $$
$$ 24V - Vce_{sat} - V_{diode}=(R1 + R3 )*I $$
$$ \frac{24V - Vce_{sat} - V_{diode}}{I}=R1 + R3 $$
$$ \frac{24V - 0.4V - 2.5V}{20mA}=R1 + R3=1055\Omega  $$

This means that to guaranty that the current will not exceed the 20mA, which is the LED limit. The sum of R1 and R3 should be greater than 1055 ohm.