Davis, the "jet needle set" as its called is also known as the "emulsifier tube". It is a long tube with a series of small holes in it that sits under the main jet. The needle that rises up and down with application of the throttle slides into it, letting more or less gas into the throat of the carburetor. The gas enters from the bowl, through the main jet, into this emulsifier tube to be atomized by the small holes in it, so that it can mix with the air coming through the throat of the carburetor. Its important that its small holes are free and clean, BUT NOT distorted or enlarged by a sharp pick in cleaning them. Same thing for the main jet, which has one small, very accurate, hole to meter the gas flow.
To dislodge and remove the jet needle set/emulsifier, try lifting the needle by turning the throttle shaft to maximum, and then without disturbing or bending the small end of the needle - push down on the brass end of the emulsifier tube that is sticking into the throat of the carburetor. It should move back up where the main jet sits. Once you confirm its loose and can slide freely, you should be able to remove it from the end where the main jet resides - it just slides out. Since that end (where the main jet rests and holds it down) is a hole without any real tolerance sensitivity, I've found I can use a small sharp screw to gently insert and jam slightly into it, allowing me to lift the tube out of the carburetor body.
One other tip! (I speak from bad experience here.) Be very careful when removing the pin that the float rocks on. They often are jammed or stuck in the post holes that hold them at each end, and if you try tapping them out then then there is a possibility of breaking a post off the main body of the carburetor.
The pin should slide out fairly easily, just by using a little leverage with a small needle nosed pliers on one end of a post. If they are gummed-up, then I would suggest a good soaking and also maybe a bit of careful heat from a heat gun or hair dryer. You need to remove the float in order to remove and properly clean the float valve, which if it doesn't seal will overflow the carbs. The body of the float valve held down with a screwed-on clamp and is also seated with another small o-ring - a different size to the one on the main jet. You can get those o-rings but the proper ones (from Honda maybe?) are quite expensive, so you might be able to find some at an auto parts supply that would be close in size. I've actually used some once that were close in size - they just needed a bit of careful trimming with a dremel sander. Just use the originals, unless they are totally disintegrated and/or torn.
