For someone with little soldering experience it's easy. I used a simple soldering iron, but without equipment it could be a bit hard 
Mine broke at an easy spot in the middle between the PCB and the cell. Depending where it breaks, it is simple to reattach with a solder blob or a short wire. Just make sure it supports enough amps throug the connection. The harder part is to open the case around the cells, especially with that glue at multiple spots without damaging the cells.
The battery is connected to the PCB with a thin sheet metal that probably broke from fatigue. I think that's commonly used technique, but it might be to weak for the very flexible VAIO case... I would have liked a more flexible connection as well.