Powering LED with 9V battery
10 mm is not an electrical specification, you should consult the datasheet to find out how much current it needs. For an indicator LED that's often 20 mA. The LED will have a voltage across it, what's called the voltage drop. We also need that, it's also in the datasheet. The voltage drop mainly depends on the color, for a red LED 2 V is a typical value.  

So with 2 V and 20 mA we can get to work. We're going to place a resistor in series with the LED to control the current. At 9 V battery voltage and a 2 V drop across the LED we'll have 7 V remaining for the resistor. Then according to Ohm's Law Voltage = Current x Resistance, we can calculate the resistor value as

\$ R =\dfrac{V}{I} = \dfrac{7 V}{20 mA} = 350 \Omega \$

The closest E12 value is 390 &Omega;. To know how long the battery will light the LED we want to know the battery's capacity, expressed in mAh, for mA-hours. An alkaline barry may have a capacity of 560 mAh. Then at 20 mA it will work for about 28 hours, maybe somewhat less.

And yes, a camera will see IR, from how far depends on the LED's power output.