I've used those pir's before they are allright for what they cost.
The output can be setup to come on and stay on for a preset amount of time on mine, not sure about the one you linked to, I got mine on ebay - it looks the same.
And I dont think it would power a high power led, the o/p is more for logic circuits or a uC, if you used something like a bc337 transistor and a 4k7 resistor you'd be able to switch an led.
I'd put led's both sides of the door.
Maybe even put a window in the door too.