Here are your parts
Replacement parts for WHIRLPOOL DU8500XX0 | AppliancePartsPros.com

Here is a wiring diagram
http://www.servicematters.com/docs/wiring/Wiring%20Sheet%20-%203370466.pdf

To troubleshoot this effectively you will need a meter.

Measure across the water valve when it should be filling and it is not.
If you have 120 volts then the valve is toast and needs replacing.
If no voltage
Measure across the overfill switch
If you have 120 volts then either the float is not actuating the switch or the switch is toast.
If no voltage across either component .
Then you either have a loose connection, a wire that is broken internally or the timer is toast.

If you do not own a meter, I would suggest you purchase a one. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Just be careful when measuring voltage.

The following is just for future reference
A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.

There is a good STICKY at the start of this forum about it's use.