Component to detect the state of an external LED
Probably a phototransistor. It would be best if you could shield it from outside light and attach it closely to the target. 

One thing to consider is if the LED is actually illuminated continuously or merely appears to be illuminated continuously. The phototransistor will likely be fast enough to pick up multiplexing that would not be obvious visually. If so then you'd need some kind of filter. 

The general approach would be to use a phototransistor plus a resistor (maybe a parallel cap) and a comparator such as an LM393 with a reference formed by another couple resistors. 

Using an LED as a photodiode (or a real photodiode) is possible, but the output current is very low compared to a phototransistor and it would thus be more sensitive to electrical noise and might require a bit more complex circuit (such as a transimpedance amplifier).