MagSafe Power Cord soldering with Power Brick
Messing with a line voltage components like transformers is a huge fire hazard. 

When these cords fail due to mechanical abrasion, it's common for other parts to melt or fray and I wouldn't trust myself to know if the repaired unit would be safe. I would worry the heat of soldering would do more damage to the cord close to the solder point. 

Have you already called Apple and had them check if that adapter can be exchanged?