Is my motor gone on my kitchen aid classic k45?

If you tip it on its side and look up its skirt you may (should) see a rating plate - a sticky label with all the important details.

My mixer says 220 - 240 Volts 50/60Hz

Your American one very probably says 110 - 120 Volts.

Changing the plug on the end of the cable does not convert for supply voltage.

There is a strong chance that you've injected 230 Volts into a 115V machine and in doing so caused considerable harm. the burning smell will have been from the motor and the motor brushes.

Harts of Stur will sell you a 230V machine for about £400

Peccavi
July 2013