If you really cannot bear to look at it one more time, ask a friend to read it over carefully. It makes a difference to produce a letter perfect document. Anything less is just sloppy work and a loss of any effort you put into the assignment. Proofreading is about discipline and pride. In a competitive workforce, your poorly crafted and sloppy letter is not going to get you the interview.
University is a time to build good habits.
One of the most important habits you will acquire, if you bother, is to proofread your work. Once you get in that frame of mind, you'll actually have to make less of an effort to proofread.