D.K.
You sound like you are doing all the right things. At 9 mos however you are right she should be able to get to sleep by herself. I would say it is time to really let her cry it out, time it, first do 5 minutes, then then next night 10 minutes and work up from there. It is hard, very hard, I know it breaks your heart but she is in desperate need of sleep and if she isn't getting it at this age that can be a bad thing.
I wouldn't allow her skipping naps either at her age she should be getting like 15 hours total, either all at night or with naps. Lack of sleep can cause all sorts of issues in babies, growth, behavior and otherwise.
Maybe find a noise machine, they have them where you can set it so if she fusses it will go off, or a music box. Both of my kids loved having one and still do.
It will take some patience to find what it will take her to soothe herself but she cannot depend on you to go to sleep and stay asleep.
For your sanity you need the sleep too, try the gradual crying it out method. I know it is hard, but if have fed her, changed her and all that then there is no reason but habit she is continuing to wake up.