A low power microcontroller — possibly a 6 or 8 pin PIC or AVR is probably your best choice for this, if you only need one input and one output.
That said, “simplest” is relative. The circuit is very simple (a chip, the LED, the battery, and the sensor– possibly with a resistor added), but the microcontroller would require programming, and you would need a programming adapter that can connect to it. (This can be as simple as extra blank locations in the circuit board, designed to mate (temporarily) to a programming adapter.)
There are other ways to do things like this as well– for example a “one-shot” (monostable multivibrator) circuit based on a low-voltage 555, such as the LMC555. No programming required, but you’ll need additional analog components to configure it, you’ll have less output headroom to drive the LED, and you’ll potentially spend more power when the LED is off.