Your concept of what the problem is is all wrong. Flashes fire almost instantaneously, like 1/20000s. The problem is the way your shutter mechanism works. It is a "focal plane shutter" meaning there are two shutter mechanisms, a top one and a bottom one. They are both up and then the top one starts moving down exposing the sensor to light then the top one starts down covering it up. The fastest that these shutters can move from top to bottom is known as the x-sync speed and is around 1/200 of a second. So to get shorter exposure, the top one starts moving down before the bottom gets to the bottom. This is why the bottom of your frame is dark. All you need to do is use a slower shutter speed or the "high speed sync" mode, which fires the flash multiple times while the shutters are moving so that the whole frame is illuminated.