The Queen's Domain sword jump, usually called Sword Flying, is a cancel. Activate the forward lunge and press Jump on the exact frame the metallic 'shing' sound begins. That timing cancels the lunge's ending ground friction, so you keep 100 percent of the dash velocity in the air and carry it into the jump. There is no double jump in the game, so this is the only way to add height and distance to a jump.
On this page
The Queen's Domain sword jump: the inputs
Equip the sword and face the gap or the target ledge.
Leave yourself a short runway so the lunge has room to build speed.
Activate the forward lunge.
Press Jump on the exact frame the 'shing' begins.
A practical tip for PC players is to rebind a second Jump command to Mouse Scroll Wheel Down. Flicking the wheel on the 'shing' cue makes chained momentum jumps far more consistent than trying to time a keypress by ear.
The early weapons guide has where the rusty sword comes from.
Which weapons work
The Rusted Short Sword has the long lunge that makes the trick work. Other weapons have much shorter lunges, so they do not carry as far: the sword you can buy in the demo shop has the same moveset but a shorter distance, and the Diamond Blade has no lunge at all.
The Holy Sword Endu fits the technique because it has both a forward dash and aerial abilities. Its Wind Dashes are added on top of the lunge, so a lunge-jump chained with a mid-air Wind Dash covers the largest gaps, and the True Holy Sword Akunra dashes ten times.
The Pandora Blade has its own version of the trick. Its secondary is a forward dash that can be chained repeatedly while you are falling, and on PC the secondary attack is right-click by default.
The weapons list has every melee weapon and its reach.
What the jump can and cannot do
The effect is horizontal reach rather than height. The lunge just launches you forwards a bit more, so the trick is useless unless you can convert that forward boost into distance, and it is what turns a short hop into a long crossing.
There is no double jump and no free climbing. Extra height comes from ledges, from the moving stone pillars, from the Cestus uppercut as an extra jump, and from the Flying Mushroom Fossil, which grants a short upward float.
Stamina is the limiting factor. Running and dashing drain the stamina meter, and an empty meter weakens your attacks, so dash in short bursts rather than holding the input down.
The Holy Sword Endu guide has the dashes the sword gives you.

Two Wind Dash trails crossing a dungeon hall; the second dash is what carries you further.