Do you want to be with her? Because "love her" and "can't get her out of my head" doesn't necessarily mean you really want to continue now. But assuming you do (because you're considering it, because you asked the question. ) read on.
Don't take her up on the cheat offer. If you do, it will make it worse and more difficult. Now BOTH of you get to feel as crappy as you do today. And since you WANT to be with your girlfriend (see the assumption above), then why bother becoming a cheater? However if you seriously want to be with someone else, why stay with the girlfriend? Rather, if you want to cheat, instead simply end it with the girlfriend and have more fun with the next girl than you would cheating. Get it?
So, now, you want your girlfriend, and have chosen not to cheat. So now you're with your dishonest girlfriend who didn't feel guilty enough to break it off, and apparently would NEVER have been trustworthy if up to her. You'd better fix that, or that will eat you up forever. Since you can't change her, all you can do is tell her how you feel. If she cares, she'll be willing to try to change herself. If she doesn't care, well, I'd rethink your decision to want her (but stick with your decision to not cheat).
So now you want her, you don't cheat, and she really cares about how you feel and what she can do to make things better. Here's what you BOTH must do: talk it through to be totally clear on why she cheated, what was wrong, and what she needs from your relationship to never feel that way again. This will take a while. Many more conversations than one.
So now you want her, you don't cheat, she cares, and you both know specifically what was wrong and what to do to fix it. One more question, that you probably got answered during your multiple fun conversations about the problem(s) in your relationship: does she REALLY want you now? If not, well, go back to square one. If so, continue to the finish line.
So you BOTH want each other, you don't cheat, she cares, you BOTH know what was wrong, you BOTH want to fix it. Now go fix it. And the whole time, you have to be nice to her (even though she cheated) and never throw it in her face and mistreat her about it. You can feel insecure or angry, but you can't BECOME hurtful or vengeful because of your insecurities or anger. Ever. Or else (and especially with this particular girl) if you do, you'll just hurting her AND be creating a brand new whole in your relationship that she's likely to cheat to fill.
So there you go. The path to forgiveness. Is she worth it?