"convince her to let me. . ." I don't think you need permission to work to gain her trust back; that's all you. But just like you can no longer be punished "proportionally" for leaving your book bag on the floor (I know that one; had all my clothes ripped out of the dresser and closet and thrown in the middle of the floor for not hanging up an item; my brother had it worse; his clothes were all thrown out of the second story window) you can practice honesty now, going forward and she may trust you again, in the future, on future subjects but if you lied, you lied. That's done and over.

Be a little less hard on yourself, you made a mistake. You are learning from your mistakes and are desirous of getting better at this honesty thing. That attitude is more helpful than beating your chest and begging forgiveness.

Of course there are bigger and smaller lies; if you chose to badly undermine the relationship and lied about that, good luck! Lying is the least of your problems. However, if you lied about leaving your book bag on the floor. . . then you both don't need forgiveness and need to learn the difference in weight of what one lies about, and how/why to "lie" in the first place because some things don't even deserve the thought of a lie.

Never choose your behavior out of what the other person is doing, thinking, might think or feel, etc. Too, know that your behavior is always yours and your "fault". It is your life. If it is your book bag, you can leave it on the floor and the other person gets to decide if they want to live with a slob or they would like to ask you to learn to pick up after yourself more often or if they are willing to pick up after you. As a child we do not have those choices because our parents are our sole support and we are "prisoners" of a sort, of them and our age/inexperience and lack of knowledge of how things work in the "real" world.

If you cheated on your girlfriend, you cheated. That is simple enough. Your girlfriend would probably have trouble with figuring out if you might cheat again, if you have changed, etc. If you lied about it; that just compounds the problem and it may/may not be the lying that bothers her more or less than the cheating but that needs discussing with her so you understand what the problem for her actually is. You would "know" your problem, you cheated and then lied. You have to decide, given all the information you can find about her thoughts and feelings and yours what you want to do next.

Knowing what you want (still not based on what she wants, her wants are just information for you, you may or may not have similar wants; she is in charge of working on getting what she wants for herself, not you, you are only responsible for yourself) you make a decision of how you want to act and that makes things a lot easier, moving forward.