Why would the water start coming out very slow?

Model number? Make?
APPLIANCE 911

You need to replace the ice maker

Two things come to mind.

First, the water valve at the bottom rear of the refrigerator may have failed. You didn't mention if there is a water dispenser and whether or not this is still working. This valve pumps water to the icemaker and water dispenser when the control unit of the icemaker calls for water. Also, there could be a blockage in the line which brings the water to the icemaker.

The other possibility is that the part of the icemaker controls which calls for water from the valve has failed. If this has happened, it would likely be more cost effective to replace the icemaker unit.

If you could observe exactly where the water is dripping from it would help but...If it overfills and drips, look for an adjustment behind the plastic endcover with a +/- on it. - is less water.
If it continuously drips, replace the water inlet valve on the back where the house water supply comes in.
The water inlet valve on the back of the unit where the water supply comes in can have a slow leak and will cause ice to buildup and block the fill tube. Disconnect the ice maker line from the valve and watch to see if it has a slow drip. It doesn’t take much to eventually block the fill tube.

Try changing the filter; our refrigerator did the same thing.

At this moment I think that you need to unplug the refrigerator and remove the ice maker to stop the leaking. Any extra water inside the freezer will make it harder for the defrost to remove the moisture. While the ice maker is out I want you to look to see if water is still leaking out the fill tube that feeds the ice maker tray. If it is then we found the reason. If not then we need to check some of the parts on the ice maker. Let me know what you find, Thanks, Sea Breeze

Make sure the fill tube going directly into the icemaker is not touching the part that receives the water, if it is touching the water will spray all over and not go into the cavity

there appears to be a problem with 1. the temperature needs to be turned down to produce Ice. also is the fill tube frozen where the water goes into the ice maker sometime you have to thaw it to get the ice out of it and the reset the maker too. Let me know if this helped and if it did not don't rate and ask another question from me and I'll be galad to help. If it did would you please rate this solution please. Thanks for using FixYa.

I have a kenmore coldspot side by side model # 106.53662300.
The ice maker red light is blinking and it has made very little ice in the last 24 hours. What do you think the problem is?

It's time to replace the dual water inlet valve (the part the water line connects to). A bit of sediment or other fine object can get in the seat or the material itself can deteriorate and allow a tiny amount of water to pass up to the icemaker.