Take the fuel tank and air filter housing off. To hell with draining the fuel it's too expensive.
Remove the metal shroud from over top the flywheel. it might look like there is more to it than just the screws you can see. But the metal screen in the middle of the flywheel shroud (cover) is usually pretty thin and you can work the shroud out from around it and not damage the metal screen. It's pretty self explanatory how to straighten it out again.
Then from the handle follow the cable it controls to the engine. It will be attached to a bracket and it controls another piece of metal that when you squeeze the handle it pushes into yet another piece of metal that touches the side of the flywheel and in turn grounds out the electrical circuit and kills the fire to the spark plug. That piece that touches the flywheel Will get stuck inside of a slot between the two magnets on the flywheel that's the break. remove the screw from the break and the break will come off that will fix it so it doesn't shut off.

In order to cut the mower off without letting it run out of gas you need to do a little redneck riggin.
First take the moving parts off all the brackets I mentioned earlier. Mainly the break it's self and the spring attached to it. Remove the arm on the bracket that the cable controls or just take the whole assembly off as long as it doesn't interfere with any other engine components. The thin piece of copper on that bracket needs to be taken off and the wire that is attached to it don't destroy the wire.

If you don't know what the coil/magneto is it is the sort of u shaped thing directly next to the flywheel and should have two wires coming out of it.
the big wire goes straight to the spark plug leave it there.
Follow the small wire from the coil to the end. make the wire longer by putting more on it try to use the same gauge wire if possible.
Then you can run it down to the mower deck and attach it somehow so that the wire doesn't ground out when you don't want it to.
Then all you have to do is a reach down and hold the wire to the mower deck till it stops running.
If you want a manual kill switch you can run that small wire to the top of the handle and attach a toggle switch to it and on the other wire on the switch run a wire back down the handle and attach it to the mower deck ( pre drill a hole and wrap the wire around a screw and tighten it).
Flip the switch and it cuts off.