Another suggestion is to get some plastic practice balls and use them for front toss with her. Get her working on that swing, swinging confidently and making contact. Then when she transitions back to real balls she may have a little less fear as well as the confidence to know that she can hit if she tries.
And to paraphrase what others have said, there's no better feeling than handing her a game ball in front of her teammates after she's just gone 0 for 20 in the first 9 games and finally got her first hit on a dribbler down third that the first baseman dropped because she never gave up... and neither did you.
I like coaches who do this sort of thing. It comes from ME having been the really clumsy kid.
I remember one time when DD 3 was on a rec league team, and was clearly the best player on the team. There was a girl on the team who had never played before, and it showed. But, she kept going through practices and scrimmages. Finally, the first game, close game. During the last inning, the coach made a last minute switch and put her at 2B. Sure enough, with the game on the line, a ball is hit right at her. She makes the catch, saves the day defensively, the team wins.
Afterwards the HC took me aside to explain why he gave that girl the game ball instead of my DD 3, whose pitching and hitting had set up the win. My reply was of course you give that girl the game ball. No brainer. That girl learned to love the game.