Thread for sewing silk?
For a 4 inch repair I would use the yarn and needle in the repair kit, and do it now before it spreads further. 
Use the most matching colour and the finest thread if you have several options, and as thin/fine a needle as you can handle for the repair.

When sewing a whole project, it is good to get good quality thread and needles, but for a small repair you use what is at hand.

While silk thread is available, it is expensive and many people will still use polyester or cotton for the hidden seams and only buy silk if there is top stitching where it shows. When you do buy it for the top stitching you might well use it for the rest of the sewing as well.