If you just want the LED to be on when the door is open, then all you need is a DC output wall-wort power supply, an LED, a resistor, and the magnetic switch.
You wire the LED in series with the resistor and connect it to the wall-wort output (positive output to the LED anode connection). The value of the resistor is determined by the desired LED current and the wall-wort output voltage. The formula is R = (Vww-Vled) / (Iled) where Vww is the wall-wort voltage (which should be at least 5V), Vled is the operating LED voltage, and Iled is the desired LED current. (The LED parameters are given on its data sheet).
Assuming the alarm magnetic switch contact is closed when the door is closed, then just connect the switch contacts to each side of the LED. That will bypass the LED current and shut it off when the contacts are closed. This will cause continuous current to flow through the resistor and switch, but it is a small amount and should not a problem.