It depends on where the wire came loose. If you have enough wire left to resolder them, you wouldn't have to replace the whole pack. If it's not enough, replace it - especially when the wire came loose exactly where it comes out of the battery pack.
Yes, the Emerald Driver is built like that. The short boards contain the solder pins and some of the electronics, the longer board contains the remaining electronics. All boards are physically connected and not held together by the tape. Otherwise the whole thing wouldn't work, actually.
Thanks! And, yes, the wire came loose exactly where it comes out of the battery pack. It kind of is laced through the Emerald Driver like so