Usually at 9 months, children are still taking 2 naps (am and pm). You haven't stated if it's one or both. You also haven't said whether or not you are checking on her when you "let her cry it out". If you are and your daughter knows you are, then you are helping to extend her crying time with your attention.
It's puzzling that she won't nap because I have never known a 9 month old baby to not need any nap time (and I've known a LOT of babies). Some babies may only nap 45 minutes a couple of times a day, some 1 to 2 hours once a day, and most, an hour in the morning and 2-3 hrs in the afternoon. My children were not big nappers, either, but they did nap.
It's also puzzling that at 9 months she is still getting a middle of the night feeding, because that should have been done with when she was 3 months old. She should be on solids (at least baby food) and shouldn't need anything more until morning. Maybe I am misinterpreting "getting up at least once".
As others have said, make sure you are providing a lot of good stimulation. Don't let her cat nap, either. I would forgo the morning nap and focus on an afternoon nap (about 30 minutes after lunch should be good since her belly will be full, and she's spent the morning wearing herself out). If she fusses, let her fuss - no checking until AFTER she has gone to sleep. I would be shocked if she cried more than 20 minutes (which I'll credit to your daughter's resistance to napping). Hopefully she'll sleep at least a couple of hours.
Watch the night time, too. She should be sleeping through till morning. Which time depends on when you put her down and what you and your husband deem acceptable for a waking time. She may be that child that doesn't sleep in (gets up before 7), but you don't have to get her just because she's awake (likewise for the nap, if you think she hasn't slept long enough). She can learn to occupy herself till you are ready for her to be up and give her breakfast. I'm not saying ignore her, but as in establishing a sleep routine, you also need to establish a waking routine. By her learning to occupy herself, it may also help for those days when she isn't ready to nap at her usual time, and knowing the routine, will figure out how to occupy herself until she does fall asleep.
Hope this helps.