How can I put double quotes inside a string within an ajax JSON response from php?

Just escape it with a backslash:

> JSON.stringify({"a": 5, "b": 'a "kitty" mighty odd'})
{"a":5,"b":"a \"kitty\" mighty odd"}
> JSON.parse('{"a":5,"b":"a \\"kitty\\" mighty odd"}')
Object
  a: 5
  b: a "kitty" mighty odd
  __proto__: Object

JSON parsers recognize \" inside double-quoted strings as a double quote. Note that in the second example, the double-backslash is needed because there’s a Javascript parser pass, then another JSON parser pass.