Which of Damon Runyon's writings are written completely using the second person point of view?
If you mean second person narrative, a quick look through my Damon Runyon short story collection "From First to Last" reveals that the vast majority of the stories in that volume are written in the first person, and all but one of the remaining stories are in the third person.
The one exception is the two-page story "No Life", which starts in the second person: "You have been noticing an uneasy sensation in region of the Darby Kelly and the croaker says it looks to him like it might be—–"
However, the rest of the story is entirely dialogue, with no tags, so this story doesn't qualify for your request for "writings ... written completely using the second person" (my emphasis).
You can search for other second-person narratives by looking through the Damon Runyon Omnibus of 47 short stories: it's a free e-book by Project Gutenberg Australia.