HawkWatch International's education team got this great question from a kindergartener in class. I have no idea. Would love to get a scientifically accurate answer back to the kid! Any insights/resources appreciated.
I saw a TV travel documentary that went to the Middle East (UAE? Dubai? I don't really recall), one of the pieces covered falconry including a bird hospital that repaired falcons' broken flight feathers. Perhaps you could contact a place like that for specific information.
Perhaps bird breeders (budgerigars, canaries etc) would have an estimate of feathers per square cm?
Feather numbers might differ depending on the type of habitat: Penguins have extremely dense feathers although quite short, while emus and ratites appear to have somewhat sparser, long feathers, but perhaps have a finer layer underneath. (Oddly, both my examples are for flightless birds.)
I had a look at some early 20th century poultry science journals but nothing stood out. Maybe some very old natural history papers might have a count?
Perhaps the answer in in your institution: taxonomists - they must have counted the number of feathers somewhere. Final idea: taxidermists?