One easy thing we see a lot is the mixer has to be in the off position when it is plugged in, or it will not turn on. It's silly but you would be shocked how often we see this. So, turn the mixer off, unplug it and plug it back in and see if it makes a difference.
If not, you are most likely looking at a bad speed control board, though you could need a new cord or motor 95 % of the time it would be a bad control board.
You can find the service manual and parts online at kitchenaidexperts.com