Is it possible?
Do you have the other end of the cable that was cut off?
- Just strip the wires and solder them back together, matching colors, and insulating each solder joint with heat shrink tubing (that can all be black - you'll see the color sticking out the end of the tube for future identification) then cover the whole kit'n'kaboodle with a big piece of heat shrink tubing to make it reasonably water tight and protect the joints. I'd suggest an added layer of electrical tape, as well around the outside to help stiffen the splice area to protect the solder joints.
You don't have the other end of the cable?
- Contact the manufacturer to order a replacement cable then see above.
Just one note: Those are extremely fine wires (18-24AWG, I'd guess) and those are not easy to work with. You also don't have much wire sticking out of that base. It might be better to open up the base and attach a factory replacement cable directly to the board inside. On the other hand, that might involve desoldering from a PC board inside and resoldering to it which could end up being even more difficult.
If you do go with soldering the individual wires, take your time and use care to not break the actual wire once you've stripped them. You don't have much extra to get a 2nd or 3rd chance with.