Note: If you like this and would want Edmund to have a happy ending just say so and I will make this a multi-chapter fic.
Edmund wanted to be wanted. He wanted to be needed and he wanted to be loved. Secretly, that is the reason why he followed the White Witch. That was why he felt compelled to go along with anything and everything she decided. He would say and do what she commanded, because she wanted him to. Even though it was for evil intentions, she wanted him around, and that's all he ever wanted.
Edmund would take love from anyone and lope after them obediently. Anyone who would take the time to understand him and care would be the one to have him. He was that desperate. He felt unappreciated and he was alone.
He would let people use him, walk all over him and abuse him, as long as he would have someone. All he needed was one person. Just one. Even if they mistreated him, because the one Edmund wanted and need didn't want or need him in return.
All he ever wanted was Peter.