What you are calling the lower part of the main jet (the part with the thread on the outside, and the screwdriver slot in the bottom) is actually the main jet. The long tube that goes directly above it, with the multitude of holes around the outside, is conventionally called the emulsion tube or emulsifier, though Americans (and Honda) tend to call it the nozzle. So, you had cleaned the emulsifier but not the main jet. You may still not have cleaned the idle jet, a black plastic object which slides down vertically into place just to the right of the idle mixture adjustment (pilot screw, in Honda-speak) and directly below the idle speed screw. The plastic idle jet has to be handled delicately, and the metering hole in the bottom of it is only 0.3 mm (12 thousandths of an inch) in diameter, so you need a fairly special object to clean it. If your engine idles perfectly it may be safer to leave it alone.
If the air filter is dry, and is not caked with dirt on the outside, it is probably good to go. Tapping it on the bench to knock loose dirt out will do no harm, but in general that method only removes caked dirt.
