You don't actually leave the level. You just have to get to the lowest point in the level to escape the blast. That means what he said, go down like it's prom night! You'll end up in a little room with some steam pipes (all I can remember). and the scene shakes around or something, and then you can go fly back up for the second half.
I recall those double doors being locked for some reason, make sure they're unlocked before dropping the disruptor.
There are four boxes to use in the defense of liberty: soap, ballot, jury, ammo. Use in that order.