This is a comic way of saying “there is no possibility/chance of” something happening. The idea is that, at first, the person had a “slim chance” — a very low possibility of something happening. But then “slim” left town — meaning, the slim chance disappeared — leaving the person with “no chance”. “None” relates to “no chance” this way: “What chance do you have?” “I have none.”

"Slim" is also a nickname for a man who is very slim (thin).

Slim = very slight chance. And no chance - well, you know what that means ;) And since "Slim just left town", that leaves only no chance" - this is a word play, perhaps, on the use of Slim as a first name. "Cakes: I have none " = "I have no cakes"