The correctly punctuated line of dialogue is: "Our lease says no pets," his mother replied.
Additionally, the line "I want a pet," James said. should also be included since it is punctuated correctly after making the adjustment to include the comma before the attribution.
The first line, "Why don't you start a dog walking business, said his sister." is missing the correct punctuation for the dialogue's end.
The line "But I really want one", he whined. should have the comma placed inside the quotation marks: "But I really want one," he whined.