As the unfavored daughter and middle child in my family...your DS has my total sympathy. She does not owe you her forgiveness and your wording here tells me that you expect her to drop her feelings because you want her to ( "She still has a chip on her shoulder" and "Look at things in light of recent events", ect"
Did you really mean to come off that way?
She does not have to accept your apology and apologizing does not wipe away the feelings that your favoritism towards her sibling inspired. Nor will it automatically fix the damaged relationship. YOU need to do the work to fix the damaged relationship and rebuild her trust. But telling her to "take a fresh objective look at the situation now to see if that was still the case and recited a few facts to back me up" just sends a message that you think she has no right to her feelings, and that she needs to change her feelings on your time table. And while she might have outwardly agreed to do as you ask...I have to wonder how she FELT about it.
She does not have to let go of what you call a chip on her shoulder just because you want it. For many people your tactic would have damaged the relationship even more.