I failed to steal from a NPC. I don't want to kill her because of a quest, but she keeps trying to kill me. What do I do?

1.) Reload. If you're a chronic saver (or if you stole from her shortly after fast traveling or entering a building) just reload your latest save (or auto-save).

2.) Run Away. Just run away. After 2-3 game days, all will be forgiven.

3.) Get caught. Just not by her. If there are any NPC 'guard' characters nearby just turn yourself in. Pay the fine, pay the bribe or serve your time. Then all is well.

In general, it takes about three days for hostile NPCs to forgive you for your actions. Try finding a bed and sleeping for a while, then come back; she will probably have forgiven you.