A coil is probably not the way to go here but I’d like to know more about the technology being used. Most appliances use membrane touchpads that are actually a bunch of switches.
However, I’ve seen a few washers that actually use capacitive buttons and it might be possible to use some foil stickers with wires connected to them to control those. I’m talking about connecting a those stickers to a ground of some sort.
In any case, this reminded me of this particular technology that I wanted to use for a product that never materialized: https://www.microchip.com/en-us/application-notes/an1626