I think Borini will start scoring again when he returns to a mid-table club in Italy, or goes to a similar level club in the Premier League. He could probably go play in the Dutch league and finish one of the top scorers there (cloggy will know better if that's true).
What often stops him from scoring is the lack of ball control and composure. He seems to know where the spaces are, and strikes the ball well enough, but he doesn't have the skill to bring the ball under control quickly. The technical skill is something he probably can't develop further at this stage, but the composure part can perhaps be fixed with regular minutes which we can't afford him.
While they included penalties (3 or 4?), he did get 12 goals in 41 games in all competitions for Sunderland last season, often playing as a wide forward. That isn't a great record, but it's still a damn sight better than the 3 in 38 for us. In Italy, he scored 10 in 26 for Roma (all competitions), in a season in which they finished 7th.
I hope Rodgers' relationship with Monk helps us in offloading him to Swansea for a decent value. They're really short of strikers there, play a similar style, and Borini should be familiar with Monk from his loan spell several seasons ago. The likes of Ki, Sigurdsson, Shelvey, Montero, Dyer and Routledge are also good/decent players, which means he isn't playing with the dross like at Sunderland. I hope Borini is reading this - get your agent to tap up Swansea!