The 'Lighter' Side Of The Force
First of all, Rey was badass with a staff so she knew what she was doing. Secondly, you said "literally" three times, none of which were necessary.
Well, Kylo was weak during the battle, but not because of the shot he took. He makes it somewhat clear that his main connection to the Dark Side is his suffering from being torn between his family and the Dark Side... When he kills his father, that doubt leaves him, his connection to the dark side diminishes and that is why he gets weaker after killing Han. He punches his shot wound constantly, to use the physical pain as a replacement, but that is not enough.