Re: Should I ask her to be my girlfriend?
I know you may not want to hear (read) this. But I don't think it wise to pursue a relationship. At least not right now.
You have to take the lead. Not sure why you are taking things slow. It's true when they say nice guys finish last. (Although the phrase is a bit vague) If you are taking things slow because you want to make sure she is comfortable then that is the wrong route to take. Women respond more to a man who takes the lead while making her feel comfortable rather than waiting around for her to decide when she's ready. A year is plenty of time to feel comfortable enough to sleep together. But it doesn't even seem like you even kissed her. Correct me if I'm wrong.
If she is more obsessed about her ex then getting into a relationship with you will likely not change that. Especially since she cheated on him. She feels guilty so that will increase the chances that she will want to correct that image she placed in his head, of her being a cheater, if he gives her another chance. Focus on developing attraction first so her focus is on you before you think about pursuing a relationship.
Also if you are not from the U.S. then obviously there will be cultural differences.
"All things are subject to interpretation. Whichever interpretation prevails at a given time is a function of power and not truth."