Hmmm.. I understand why she may be angry with you doing that behind her back, pretending to be her ex. But within time, she will forgive you because right now her emotions have taken over (anger) so she cannot make decisions to forgive you based on her anger. Let’s see her response with the letter and let me know if you’d like further help. There’s only so much you could do, begging won’t help, you can only show someone how you truly feel and leave it as that but do not over do it.