As, you said after trying many suggestions, there is no magical formula that works in this situation, but only your honesty and intuition. Use them wisely with emotional tones that break the iceberg in her heart. Words like these could work: There is no life for me without you coloring it by your kindness and forgiveness. I made a silly mistake and I am regretting. Could you allow your mind to destroy such beautiful relation?
Being a kinda unserious person is not OK. I am referring to the use of her pc to look for a job and then get into the mode of looking for girls. Ask yourself honestly, why you did that! Was it because she treated you badly and you wanted to get into some other affairs?
It happens sometimes to some people, but that shouldn't make their earth painful. Everybody makes a mistake, or some and they apologize and get their apologies accepted. So, why should yours be refused?
It is good to be sure if she is dating, or not before doing anything else. Even if she is dating, you can ask her for a meeting to tell her not about how much you love her, but about facts that you have never cheated on her. Admit that you did a mistake while using her computer and ask her is that a big deal!
Tell her that even when people make mistakes (and you haven't) forgiveness is created to make their lives better. Tell her that you tried to walk away, but you couldn't, as every soft breeze reminds you of her, every sunset, or sunrise reminds you of her and such emotional expressions. Be always serious and in good temper.
Could you afford giving her the gift of the most desirable thing that she likes to have?
You said, "I sent her a letter and poured my heart out". What did you write in that letter? It is important in such matter (when you write) to write about solid facts that you love her, you have never cheated on her and remind her of her last words about friendship. Admit that you did a foolish thing and let her know that will never happen again. Again, while saying that you accept her proposal for being friends, assure her that you are serious about everything that matters with her.
But, sometimes doing such things may not work. Getting the job could solve many problems, as it doesn't only show seriousness, but independency and determination to make oneself better. It also shows that you will never depend on her for living.
You'll get some useful gifts when you do. The gifts are good to set up with your passion, so you could convert it into a business of your own on solid grounds, even if you are a student.
Then, you'll find that you don't need to work for a company, or someone else when you graduate.