in the quest the horn of Jorgen Windcaller, you are sent to get the horn for the greybeards, however it has already been taken by Delphine but i have to wonder how she got past the gates that require Whirlwind sprint to get past to even reach the horn.

anyone care to enlighten me on that matter?

My guess is she managed to pry the backdoor open somehow.

This seems logical to me, as there are dead Draugr in that room that she most likely killed.

She's got plot powers. She uses them to advance the story at all costs, regardless of how much or how little sense her actions make.

She used Instant Transmission and appeared right at the horn. Suddenly, two Draugr attacked. Having experience as a Blade, she made short work of them. She took the horn, left the reward for us, used the secret entrance and escaped.

Oh she probably just yelled at the door loud enough so it was like "fine [censored]" and opened

i lost my [censored] a Instant Transmission

This is a part of the story that makes no logical sense. The developers obviously didn't think it over very well, since the Dragonborn is the only one (besides the greybeards) that knows Whirlwind Sprint.