the problem will more then lilkley just be the fact that it needs cleaning inside the transmission cover,the return spring and small intermediate kickstart gear get covered in rubber dust from the wearing drive belt and jam up,simply take the cover off (leaving the lever in situ)and clean the kickstart quadrant and spring and gear with petrol and a brush dry it off and lubricate it all with some light oil like 3 in one.work it up and down until its free again and refit it.this is a very common problem with auto scooters.check the gears are not damaged when its apart and if it is then replace any damaged parts.
adam bike doctor, June 2010