For a circuit, the simplest way would be your sensor, a battery a dropping resistor and the LED's,
The more correct way would be to use a constant current boost converter after the sensor, most LED light strips have a little module that can be pinched, as this will get you more consistant brightness and a bit more time for the same capacity batteries,
as for your battery life, that comes down to how much power you are putting through the LED's, as a rule of thumb to make more sense to you, if you are pushing 20mA through your LEDs in series, for a basic alkaline battery your looking at about 2000mAH (you will need enough to make its voltage exceed the LED's for the simple circuit), thus for that you can expect about 100 Hours before it goes flat,
If you used cheap rechargables (~700mAH) you would expect about 35 hours before its flat, but the perk would be you could expose a metal pad in 2 of the finger holes to charge it back up,