[An only parent, even a mother, CAN effectively parent and discipline. And I had a very difficult child.]
I agree, and I've seen quite a few examples of well behaved and respectful children raised by single parents, as well as spoiled, ill-behaved children in traditional households.
But the task of instilling discipline in children is inherently more difficult with only one proper authority figure (no matter how consistently good at parenting) than with two present in the same home. When my wife departed in 2001, my daughter became "a very difficult child" as well. It's taken a whole lot of extra effort to keep her on the right path, essentially having to do all the duties of two parents, myself. From what you wrote, I can guess that you're as familiar as I am (and with so many others, now) with just how much harder it is to be both the mom and the dad at the same time.
Now that my daughter is thirteen, I find that I'm inadequate as the female role model that she needs (and badly misses) but I don't know how to provide that for her at this time. In a similar way, teenage boys need a strong male role model living with them to demonstrate, by example, how a proper man should be strong and robust while always practicing restraint and self discipline.
Get her involved in a purposeful activity with a strong female leader with whose values you agree, and who will reinforce what you teach at home, and you will be just fine.
Additionally, at her age she needs a positive activity other than hanging out with other girls, spending time at the mall and getting led into inappropriate behavior. An engrossing hobby/sport at her age and for the next few years will save you a lot of heartache and do her a world of good.
Another way to handle it is take her a lot of places with you, talk to her a lot, do a lot of listening, and let her see what the adult world is all about. Many times it is an effective antidote to the teenage poison peer group.
You're right. It's very hard to be a single parent. And your daughter is at a tough age. By the time my daughter was 13, I was married. And I had the added advantage that after at least 10 years of the terrible twos, her worst was over. She actually settled down to be a decent teen-ager. Just stay loving, tough and consistent and remember that it often takes years to see the results. Teens need lots of boundaries--careful, thoughtful ones--although most of them would rather die than admit that if they realize it at all. I also tried to negotiate on issues that weren't as important and that gave her the impression that I was reasonable. I tried to be, but wasn't always. Hange in there, you've got a difficult task!