Long broken is hard to do. Me and a freind stopped talking now less than probably 10 times. Every time it ended I thought, "This is it. It's really over." I would send a letter apologizing, whether it was my fault or not. A lot of time times, that would not work immediately. But all of a sudden, I'd get contacted again when I least expected it.
To be fair, that person had no other friends, and kind of needed me.
I would just write an email. Talking about the reason your friendship broke. Take accountability if it was your fault. Apologize. Be honest. Tell them, that you miss their friendship. Then leave the rest up to fate.
I will say, I've learned that friendships are not everlasting. People change. Things happen. Sometimes your meant to be friends with someone for a period of time. Then your meant to move on, having learned, and grown from that friendship.