She clearly cares about you. You need to focus on yourself. She still seems to be your friend and trying to support you. Don't read anything into that, however people's feelings rarely change overnight.
IMHO you need not to push her. Don't withhold knowledge of the surgery from her, she will feel not trusted. Let her know you are going to work on yourself and your anger issues. When you have yourself under control you will be able to have a relationship, until then you are not in a fit state to ask someone else to be involved with you. There are reasons behind your actions, you need to understand the root cause of your anger and control issues, you need to deal with those and take measures to change your reaction to situations, so you can be calm and relaxed, which will mean she will have no cause to be afraid of you.
Then and only then can you look at asking her for a second chance. To be fair she may stay supportive and be a good carer and friend in this period. So she will notice what you have done anyway.
I'm not religious but this resonates with me from the Bible: Corinthians 13:4 to 13:7 "4 Love is patient, love is kind. It does not envy, it does not boast, it is not proud. 5 It does not dishonour others, it is not self-seeking, it is not easily angered, it keeps no record of wrongs. 6 Love does not delight in evil but rejoices with the truth. 7 It always protects, always trusts, always hopes, always perseveres."