Connecting a light to a battery
A LED will light up at its forward voltage, which is 2.1-3V depending on its color. Also keep in mind that LEDs are sensitive to polarity: simply swapping the two pins of the LED in your circuit may work.

However, please consider some generic comments:


you won't be able to light up the 120V night light when connecting batteries to its plug, since it needs 120V, obviously :) 
LEDs - unlike bulbs - require constant current power. It means you need to have a resistor in the circuit, otherwise 1.5V battery won't be enough to light up the LED, but 3V will fry a red LED. Get any resistor at a value of 120-220 ohm, also 470 ohm will work at 4.5V. (Comment: the 3 LR41 case worked without a resistor, since these small batteries have a high internal resistance - although it makes 4.5V, it won't fry the LED but the LED will "force" the battery to its forward voltage.)
LEDs are sensitive to polarity: they work when their proper side goes to the + terminal of the battery. If does not work, just swap the two pins of the LED.


Said that, it is a lot easier to try with a small light bulb, from a torch.
Even a small spare bulb from your car could work: those are rated for 12V, but produce light at 4.5V too (use 3x C or D cells). Pick the (physically) smallest bulb from the replacement kit (the big headlight bulbs as well as the indicator bulbs need more power than a small battery can provide conveniently - although the D batteries will produce light.)

If you are still for the LED, please:


use the 3x LR41 battery setup to verify if your LED works at all. (You might have fry that with 3x D already)
Note which pin of the LED connects to the + side marked of the LR41
use 2x C, connect these in a way that + of one battery goes to the - of the another battery. Now you have the - of the first battery, and + of the second battery unconnected, forming a 3V battery bank.
connect the + end of the battery bank to the resistor, the other end of the resistor to the + side of the LED (learned with the LR41)
connect the other side of the LED to the - side of battery bank


LED must work. 
If you do not have a resistor you can still give it a try, however be prepared the risk that LED might fry sooner or later. (It will not blow up or anything, just it won't work any more.). Red and yellow LEDs are working well at 2.1V, green works around 3V, most white works around 4V (although officially you shall supply about 20mA current trough the LED to drive it properly - and the resistor is an easy way to do that).