I sense a problem with order of operations here... did you cheat on her before a month ago when you were together... or did you sleep with someone she hates after you broke up with her and she just didn't like that?
In regards to your actions towards her.. it's as Carl said... you have to decide if you want to be with her or not (and no... as a friend is not an option at this time).
If you want to be with her then take things slow, while still letting it be known that you do want to get back together.
If you do not want to be with her then you need to see less and less of her.. eventually cutting most if not all ties with her. Do this for a few months (possibly longer depending on how long it takes for her to heal)... then it may be possible to resume a friendship (if she wants to by then).
You can't have both --- her having feelings for you and being a friend as well. That is cruel, counterproductive.... and will put her through unnecessary pain for purely selfish reasons on your part.
"The weakest soul, knowing its own weakness, and believing this truth that strength can only be developed by effort and practice, will, thus believing, at once begin to exert itself, and, adding effort to effort, patience to patience, and strength to strength, will never cease to develop, and will at last grow divinely strong."
- James Allen