Hi, my name is Mike and I have been in the home improvement and repair field for 20 years. I will be very glad to help you today.
If you look behind the piece that you push the cup against to get water out there is a little button there that may be sticking. You may be able to lubricate that button. Only use silicone spray or water proof grease. Do not use any petroleum products or WD-40. Let me know if that solves the issue. If it does not we can certainly continue to communicate until the issue is resolved.
If you turned off the water supply then you will not have ice either. Did you try what I suggested to fix the leaking water dispenser?
If you do not feel that the button is sticking then the water valve is failing. The water valve will have to be replaced.
What is the model number of your fridge?
You would first order the replacement water valve. Access the back of the fridge and you will find where the water connects to the fridge and it is connected to the water valve. Turn the water off and disconnect the water. Turn off the power. Remove the bottom, back panel to access the valve. Take note of where the wires are connected to the valve. Remove the screws that anchor the valve. To disconnect the quick connect water lines, push the plastic sleeve in toward the connection and once the sleeve has pushed in and released the teeth, pull the water line out. take the wires off of the old valve one at a time and connect them to the new valve, connect the water lines and anchor the new valve.
I am unsure of the cost. The best way to go about it as far as ordering is to find a local appliance parts store and order from there. Have the model number and serial number of the fridge and they can order the correct part. I can say that it won't be a lot. If I were to guess I would say about £25
I haven't heard from you. Was the information I provided helpful?