You can order a new cable from one of the sources mentioned here are you can take the old one with you and go to a local bike shop and get a cable from them. You will have to file the button down (on the carb end) to make it the same size as the original.
The reassembly sequence is as follows.
put the "pacman" looking thingy (the needle valve clip) on the pointy thing (needle valve), second notch from the top.
Drop this down inside the big open end of the cylindrical silver piece (throttle slide) so the needle sticks out the bottom.
Now drop in the Peace Symbol washer and rotate it and the pacman thingy so they are properly seated at the bottom of the throttle slide.
Now insert the coil spring inside the throttle slide. Assuming you have the new cable threaded up through the carb top, elbow pipe ans cable sheath with the button end down on the carb end, now thread the cable down through the coil spring and slip it in the slip notch and slide it over so it locks.
Sometimes it helps to turn the assembly upside down and push the cable up while compressing the spring so you can lock the cable button in place.
I am assuming your throttle handle has a slider mechanism where the plain end of the throttle cable threads through a hole in the handle and then into the slider piece. You have to loosen the very small screw on the slider piece so the cable can thread through. Once it is through, pull it all the way (you can cut the excess off later). Now tighten the securing screw and twist the handle. You should see the spring compress in the carb throttle assembly.
Now put the throttle slide assembly and carb top back on. Look at the throttle slide, you will a vertical groove in one side. Look inside the carb casing where the slide is to be inserted and you should see a tiny protrusion that the groove has to line up on. You should have a flat metal piece that hooks to the bottom of the choke stem, under the carb top.
Slide these parts back in the carb and fasten down the top (don't over tighten) and then push down on the choke stem (tall vertical stem in carb top. Now if you twist the throttle handle, the choke stem should come up.
You can start the ped and adjust the idle using the adjustor screw on the elbow pipe where the cable housing meets the elbow pipe. You can also do final idle fine tune with the screw with the spring around it on the outside of the carb. Screw in to idle faster, out for slower.
When you think you have everything working properly, now you can cut off the excess throttle cable at the throttle handle.