Attempting to switch on and off a 4.5v solenoid with a transistor. using push button to control input
I don't see any resistors in your circuit, only a transistor in the TO-92. So probably your transistor is MOSFET and its Ron is too high. 
 Try another transistor with lower Ron (1..50mΩ).
 LED requires just few mA to illuminate unlike the solenoid which requires much higher current to turn on.