Not familiar with the foot switch (mine only had a water pump bulb to wet the screen then I think you had to manually start the wipers) but if it is automatically starting the wipers AND the water pump then it would be a momentary switch that grounds the low speed circuit thus activating the wipers. The existing self-park system would then take care of returning the wipers to park once the foot switch is released.
Either pick yourself up a replacement wiper motor or take your apart and clean the internal contacts. You may even have a broken wire inside.
Ignore the photo of my wiper motor. I was going to tell you what each wire did but I cannot find my diagrams. Also, nobody else ever had the same colour wires as me!!
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