To whom does she tender the suit for a defense? Duke? Her department chair? Her renter's insurance?
If you worked for NC State (as I did), there is something called "The Defense of State Employees Act" which permits the state to defend you in the performance of your duties. The key word there is "permits". I found this out because they were forcing me to write and sign a lab safety plan and threatened me with the loss of my lab if I didn't do it. I knew good and well that if anyone ever got hurt in that lab they would throw me over the side and I'd be stuck.
I don't know what private universities do.