WARNING! Before attempting to work on any appliance, make sure that all power (electricity) and utilities (water and gas) have been turned off and/or disconnected on the appliance. Read this before you start any repair.
Turn off the water and disconnect the fill hoses from the back of the washer where the washer hoses connected. You can try cleaning the fill valve screens on the hoses (if they have them) before hooking them up and seeing if this improves how much and fast water fills the washing machine.
If this does not solve the problem, then the problem might lie with a defective water inlet valve. A washing machine water inlet valve also has screens that catch sediment where it connects to the washer hoses. You can try cleaning these as well.
The water inlet valve controls the flow of water into the washer through solenoids. Over time, the water inlet valve can break causing there to be no water, low water pressure, water not shutting off/ entering tub when machine is off, leaks and overfills.
Other problems causing a slow-filling washing machine could be low water pressure from your home or the washer lid switch.
Find washer water inlet valves: http://www.everyappliancepart.com/Category/Washer-Parts/Water-Inlet-Valves
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