First off you got the right chip.
Next if you simply connected the input to the anode of an LED ( and connected the ground to the cathode ), there is not enough voltage to turn on the logic gate and so the output will always be high. This assumes the current limiting resistor is in the anode of the LED. If your "thing" is a 5 V system then connect your gate input to the other end of the LED's resistor.
While all inputs should be connected to something, it is not something that will stop the gate working. In fact I have never had problems with this when testing circuits.
Having said that then a transistor would be simpler.