In case anyone genuinely has this misconception: birds branched off from the other dinosaurs during the Jurassic, probably over 100 million years before the astroid hit. Dinos didn’t suddenly grow feathers and a beak because a big rock hit them.
It seems quite a few modern birds (Aves) lineages survived the K-Pg extinction (at least 5, last I checked), but when exactly they diversified is apparently still a contentious issue. The common ancestor almost definitely lived sometime during the cretaceous, so not THAT long ago in the grand scheme of things, but it definitely lived either before or during T-rex’s reign.
I was referring to Avialae, which is the clade defined as all dinosaurs more closely related to budgies than to deinonychus. Many of them would have seemed quite birdy to us, but like the other dinosaurs not many of them made it to the current day and the ones that did are all Aves.