So I see a blue button is supposed to drop the bridge to cross the gap. But the button seems to not be working and this my first time in this location so it's not the known bug of when you return and the button won't function if the bridge is up again.

Will the Whirlwind sprint shout work here or am I totally SOL?

Thanks!!

Mechanical failure caused by wedging of objects into mechanisms is your problem
Look around at the gears.