I know this is old but my solution may work for others.
What I ended up doing is striping the broken wire back a bit, twisting the exposed wires together, and inserting it into the hole it came out of on the connector. The wire will go in a few millimeters. After this I used superglue ( the kind that has a small jar with a brush) to secure the wire to the connector on the outside. I also used the super glue to cover up any exposed wire and on the other wire too to keep it from breaking in the future.
Just be sure to let it dry and not get an on the connector itself. It could keep it from plugging into the logic board.