For dunked stuff it's best to take any power sources out and keep wet. Then dissasemble as much as is sensible wash the bits under clean running water dry off with something absorbent then give it the warm place for a day or two treatment.
I think for your needs a small lightly ventilated box so you can bring the whole object up to 30 or 40C to dry would be better than just a warm mat that will really only heat the bits it's touching. As this needs to be warm for a day or three or longer some sort of powered heat source rather than a Hottie or hotwater bottle is required. We are only talking a small box with not a great deal of ventilation, so I suspect a 21W 12V brake light bulb would do, if that gets too warm try a tail light bulb.