The nurse at the end is reading a paperback version of the crime novel and opens the rear cover to look at the photo of Kara. Paperbacks do not have the author's photo inside the cover, only hardback books do so.
Charlie had the hardback edition of "Murder in the First Person", telling Kara it was a good book, indicating she had not only read it but liked it. As such she would have seen the author's photo inside the dust cover and recognized Kara.
Kara finds out after talking to "Paul" that Sheldon & Johnson were his publishers and acts surprised. In the rejection letter she received, it noted that they had published "Paul's" book "Murder in the First Person", and additionally, she had a copy of that book where she was staying, so she would have known who published it.
Jason was a law student. As such, he was not admitted to the bar and could not be Kara's legal counsel.
Kara gets a text that Charlie won't be home due to a hot date. Less than a minute later she asks if Jason would like to come inside until Charlie gets back. Since it was not a sexual invite to spend the night, her line did not make sense.