Just to recap: A flat character is one that is not really fully 'fleshed' out as far as information, development, or personality. They are usually secondary characters. A round character may be a secondary character, but as readers we are privy to more information and internal musings.
John Oakhurst: Professional gambler who is forced out of the town of Poker Flat as an undesirable because he wins so much money from the townspeople. Definitely round.
The Duchess and Mother Shipton: Prostitutes who are also banished from Poker Flat. Flat
Uncle Billy: Thief and drunkard who is likewise forced out of town. Flat
Tom Simson: Young man who leaves town and joins Oakhurst and the other outcasts. He admires Oakhurst, who returned $40 he had won from Billy after feeling sorry for him. Round.
Piney Woods: Tom's sweetheart, who leaves with him. They had planned to marry in Poker Flat. Flat
Jim Wheeler: Resident of Poker Flat who bemoans the loss of money to Oakhurst.
Jake Woods: Piney's father. He is referred to in the story but plays no active role in it. Flat