If the choke is functioning properly then you'll probably need to go through the carburetor and replace the diaphragm. This engine is shown to have been available with two different carburetors, one with the diaphragm on the side and the other with it between the tank and carb. Both of these are very simple carburetors and you should be able to take care of them with a few tools and some spray carb cleaner.
It sounds as if the diaphragm may be having trouble picking up fuel at the low engine speed you get when trying to start it.
You should also try loosening the gas cap and try starting it and replacing the spark plug but I think the diaphragm is going to be the culprit from what you describe.