‘Dats a nice hoverboard. Electric start (only) without retaining the older kick starter pedal as a backup to battery, starter or starting circuit failure leaves you or your repair shop to diagnose where the problem(s) lie(s). If this is a 12v battery, can you boost it from a car battery? Does the battery measure 12v from a multimeter? An inexpensive trickle charger outputting less than 2 amps is more than sufficient to recharge small 12v batteries. Unfortunately, motorcycle batteries have a shorter life than car batteries unless they’re sealed gel or AGM types to last longer.
First thing to try is starting from a car battery if possible. Your hoverboard won’t need too much power for the starter but try a battery boost to see if starting occurs. If the starter runs then the engine should startup unless other issues prevents the engine from running. If the starter doesn’t engage to crank the engine then the problem may be wiring, starter button or starter. An alternate method would be a running start, preferably going downhill to use gravity to get the bike up above, say 15-20 mph and letting the clutch out in high gear to turn the engine over. The battery should have at least 10 volts to run the electronics for the ignition system. A running start eliminates the starter and can tell you if the engine runs. The battery must have sufficient voltage to power the electrical system. A running start is no different from pushing a car with a manual xmission to startup in gear with a partially dead battery.