What would you do and how would you do it.?

ok...met a gal in april who told me upfront her mom is an alcoholic and her parents have been split since she was one.....dated for two months....was telling me we were exclusive..but I could tell she wasn't 100% honest....caught her in a few lies.....gave her a sec chance....then we had a fight and I went over to makeup and she was in bed with another guy......gave her the papers......month goes by her friends call..she stops over...etc.....begs for forgiveness and blames it on her lack of self esteem and her upbringing......I slowly feel sorry..listen and slowly we get back together...she has been great for past two months...but........I know that underlying issue does not get fixed overnight...so despite her going to therapy, church, and Alc meetings to understand.......I just can't get past the betrayal and cheating.....so......how do I tactfully end this thing with a girl I care about and feel sorry for........