Lets get some terms straight so that we don't all get confused.
The carburetor has a "float" that operates a valve the controls the level of gas in the "bowl". The thottle cable does not attach to the float in any way.
The typical two stroke carburetor has the throttle cable pass through the top of the carburetor and attach directly to the throttle valve, which is a rectangular "slide" that moves up and down. In the process of attaching to the slide it also connects to a "metering needle" that moves up and down with the slide. The metering needle will have several options for where it connects and that is used to adjust the amount the mixture is changed over the throttle range.
You should be able to remove the two screws that hold the top cap of the carburetor off and pull the throttle slide & metering needle out. Twisting the throttle should pull on the cable, pulling the slide up towards the "cap", compressing the spring. If that doesn't happen then something is wrong with the cable.
If the throttle cable is okay then it is possible that you had something mess up the carburetor. Perhaps some water got in the tank, or some dirt, and it worked its way into the carb. It could have plugged the float valve so that gas isn't entering the carburetor or it could have plugged up the main jet inside.
If you take the entire carburetor off and then turn it upside down you should have abotu a 1/2 cup of gas spill out. If there was no gas in the carburetor then either the float valve was plugged up or the tank was plugged up.
Properly cleaning the carburetor is a bit more involve than just taking it half apart and wiping it down. To really get the important areas clean you need to remove the main jet, float valve, idle screw, idle jet, etc. to open all the passages. Soak all the metal parts in a really strong solvent (carburetor cleaner) overnight. Wash everything in clean water, blow dry with compressed air. Blow compressed air through all the little passages to clean them out. You really should use all new gaskets to reassemble, and a new float valve seat is a good idea. The float level will need to be checked and adjusted anytime you have taken the carburetor apart, especially if you have removed the float valve (which you must do before soaking the carb in solvent, you don't want to soak the valve).
Before you start tearing the carburetor apart you should make sure that you are getting spark.
Rod