Simple MOSFET circuit doesn't seem to be fully turning on
The problem is as you turn on the MOSFET, current flows through the solenoid and the MOSFET source voltage rises and tries to turn off the MOSFET. This is negative feedback and will force the MOSFET to behave linearly rather than act as a switch. It would be better if you moved the solenoid to be in series with the drain connection on the MOSFET: -



If you want the LED to extinguish when the button is pressed move the LED and series resistor to connect between drain node and source node of the MOSFET.

On the other hand if you want the LED to light when the button is pressed, then reconnect the resistor and LED across the 10 kohm resistor.

Breaking news: -


  Unfortunately, the LED and button in the button housing outside are
  connected to the bell, power supply and mosfet by only a 2-core cable


Then you will have to use a low value of resistor instead of the 10 kohm in the diagram above.