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Replacement parts for KENMORE 11085872401 by Kenmore Elite | AppliancePartsPros.com

I beleive your unit is a close match to the following.
https://secured.whirlpool.com/Service/SrvTechAdm.nsf/2cd44500d572193285256a45004fd9d6/cdce3c10dcb5382285256aa20064beaa/$FILE/Duetdry.pdf

I am assuming that everything looks normal (interior light, control panel etc.) but it just will not start.

You could open up the unit and check it with a meter, see page 4-9 in the manual.
Who knows you may just have a broken belt.
OR
You could unplug the unit, disconnect P1 from the board and then with the door open measure across the plug from P1-4 to P1-3 for continuity.
This will check the thermal fuse, the belt switch and the motor windings.
If this is infinite ohms (open) then you can dig further.

I would definitely complain about the service you received as it sounds like the service persons poor skills have cost you a lot of money.

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.
Most places will not let you return electrical parts so if you order it, you own it.
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.