How do I clean a really old kitchen appliance?

Anyone ever had an appliance that got that sticky stuff on it from age?
My mom gave me two appliances, one a sandwich maker thing, and another a waffle iron I remember using while growing up. I tried to clean them, but they have this sticky stuff on them. I tried using several products but nothing gets it off, and it even takes some scrubbing to get it off my hands after holding them. Just like really old kitchen grease or something. I plugged the sandwich one on, and the smell was awful; it just smelt just nasty. I cannot get them too wet, as the electric coil and such cannot get wet.
They both work very well, and I do not want to buy new ones. I just need to clean them up before using them. Any ideas?