I will email her and tell you how she responds. This is a good example of business speak, sadly.

To ping someone is to check with them for some response. It could be in the form of email, IM, or some other electronic mechanism. To circle back is just to come back to let the other people in the conversation know what the response was.

"I will ping you the code" - Ping is used to mean "message" (typically via something like Skype, workchat, etc.) "Let's circle back to the first topic" - Circle is used to mean "return"