I hear too much drama over a few words "bird strike" and not enough emphasis on doing what needs to be done. Sure take a look at your instruments "not much more than you should be doing anyways", do not let it distract you from flying your aircraft and if by chance your SOP says DECLARE AN EMERGENCY because of tweety bouncing off your screen it's ATB time.
I recovered a classic 742 with JT9D-A's or J's on her and found easily 30% of #3 engines fan blades missing and teradyctle bones sticking out of the nose cowl. Boys in the cockpit had nothing in the book. Did remember a slight vibration @T/O thrust but it went away as they retarded to climb settings.
Old boy Sully lands one in the drink and the fever starts over the result of a freak accident, with no human loss.