I presume you have a soldering iron. If not, go buy a cheap one. they are not very expensive and will pay for it self in repairs. I also suggest heat shrink, but electrical tape works too. Now yes, those wires are VERY tiny and delicate, but you can do it. Here is a vid on how to splice wires. and this goes for any wires.
https://www.youtube.com/watch?v=O-ymw7d_nYo
Best of luck! And try not to pull out your hair when you break the cable in the process XD