This sounds really tough!
Sometimes even people who love each other still won't be compatible in all areas of a relationship. Though it is often assumed that sex shouldn't be such a big deal, that isn't really a fair assumption and if people are really sexually incompatible, then they might not be the right people for each other.
There are a lot of red flags here. The fact that you cheated to find a sex partner, the fact you want to have sex and your girlfriend doesn't and the fact that she won't talk about the issue are all signs that there are real problems in the relationship.
Unfortunately, unless you're a mind reader you can only guess why your girlfriend doesn't want to have sex. Here are a few issues that could be contributing:
- She might not feel ready
- She might be questioning her sexual orientation and who she is attracted to
- She might have been a victim of sexual abuse
- She might have guilt and shame about sex
- She might be worried about her performance
- She might have specific needs she doesn't know how to express
- She might be on medication, like anti-depressants, that affect her sex drive
Of course, it could be something else altogether. But if she is being tight lipped, there's no way for you to actually get to the heart of the situation.
The way you describe the situation, it doesn't sounds like either of you are getting what you need from a partner. Ending a relationship can be really hard and scary, yet sometimes it is the best thing for both partners. But if it seems to difficult to break up immediately, you can always think about talking to a counselor, with or without your girlfriend.