What to do when you called your mom a b*tch "not really saying it to her but writing it in a letter" and she found out? How do you restore the bond when it's already broken? How do you forgive yourself?

She must have read what you wrote down when you were mad at her. Just go to her and apologize. You're sorry that you said that in the heat of the moment and your emotions got the better of you. Next time, just think it, instead of writing or saying it.

Frankly, she's overreacting.  I suspect you reacted with anger to something she said or did or did not let you do.  You tried to defuse your anger by by writing a letter that was never intended to be sent. That was a good idea.  Fully functioning adults sometimes use that technique. Unfortunately, she found out.

Tell her you were very angry when whatever happened happened. You expressed your anger in a word you considered powerful to sum up your feelings.  You had a normal human response to what happened. 

No bond has been broken.  And I don't think you need to forgive yourself for responding like a human being---anger can be a valid, appropriate response.

You can apologize for using a word that she was never supposed to find out about because it probably hurt her, but you don't have to apologize for being angry or upset about what she said or did.

Now she needs to act like an adult and help you deal with the whole situation.