AtD translation: how deeply of your persuasion?

[Mike Jing]

P861.23-32   “I saw you with someone, didn’t I, over on the Lido? Only a
glimpse in all the confusion, but she seemed quite appealing. Actually.”
       “To you?” Careful not to seem too puzzled and provoke anything too
soon.
       Theign shrugged. “More to the point, how appealing to you? And how
deeply of your persuasion? Or would she’ve been there more in a decorative
role?”
       “Are you asking, what would I consider trading her for?”
       “Of course that does happen now and then. But I did not wish to
insult either of you.”
       “I don’t know where she is. Even if I did, she’d be of little
use. . . .”

What does "how deeply of your persuasion?" mean?