Here is the illustrated parts list:
http://bsintek.basco.com/BriggsDocumentDisplay/Z6kotCVJ1DajI.pdf

Here is the operator's manual:
http://bsintek.basco.com/BriggsDocumentDisplay/msowGK-K_ny7tH217FzoPu.pdf

A couple of points to start with. First, never try to run a VacuJet carburetor like yours, or especially a PulsaJet carburetor, without inserting the air cleaner retaining screw and screwing it in. Second, you do not seem to have followed an orderly process with your attempts to get it running, and this might have made your efforts futile. For example, it is very easy to make a 4 stroke too rich to run, and with the air cleaner screw not installed plus a random quantity of fuel poured into the carburetor air intake, you could easily have exceeded this limit.

I suggest you switch to an orderly process instead of a disorderly one. First, prepare the machine by inserting the air cleaner screw and setting the speed control in the start position. Remove the spark plug and pull the starter half a dozen times to begin the process of clearing out surplus fuel, if it has any.
Second, test for spark by connecting the plug wire to the plug (which has been removed from the cylinder head) and clamping the metal plug body to the cylinder head. Pull the starter and watch for a steady series of blue sparks across the plug gap. If you don't get this steady series, stop right there and fix the ignition system. If you do get satisfactory spark, reinstall the spark plug and connect it.
Third, you need to check whether the automatic choke is functioning correctly. Look down the bore of the carburetor and observe whether the choke is fully closed. If it is, pull the starter and see if it opens while the engine spins, then closes again as the engine stops spinning. If the choke isn't fully closed, you've wrecked it and will have to go through the carburetor overhaul procedure from the Briggs manual.
Fourth, check the compression. This means first, feel the resistance to pulling the starter. Second, disconnect the spark plug lead and tilt the mower so you can access the blade disk. Now check two or three more times that the spark plug is disconnected, before you go any further. Move the speed control to the stop position. Rotate the blade disk backwards until you find compression, approaching it from the wrong side. Then flip the disk backward toward compression, and see if it bounces back.

If you have functional spark, compression and choke, you are ready to start finding and fixing the fault. Please report back at that point.