A simple finger trap splice is as strong as you could want and very easy to do. See the videos above. Most after market line is much stronger than the stuff provided with your kite.
Simply make plain lines with spliced loop ends. You can sleeve the loop ends if you want. I double splice them (insert a length of line into the loop end and part way down the bury).
You will want to sew the bury to stop the splices slipping. You can do it in a domestic sewing machine or do it by hand.
For the pig tails use 3mm dyneema (or equivalent). It is ridiculously strong and easy to splice and sew.
Tip. Splice and sew the first line. Make the second line and adjust it to length and put a safety pin to lock the splice. That way you can get the lengths identical and take it to the sewing machine without anything slipping.
I buy line from
ropegalore.com.au/dyneema-r-high-strength-rope/ I used to get line from a guy in Holland but he doesn't seem to be in business any more.
Tip 2. Splicing is easy to do but it takes all day to mark it and cut it and splice it and sew it and all the other dicking around. (That's what I am doing today so I know.)
PS. I just finished making a line set. If you're careful with the marking out then it's easy to make identical length lines. The main thing to get right is the centre point of the loop. I also mark out both ends of the loop and the end point of the bury.
PPS. If it was me I'd shorten and splice loops into the broken line and its pair then add 3mm dyneema pig tails.
PPPS. From your picture above the kite maker has simply looped the end of the line and used it as an attachment point. That's crap. You want a loop-loop attachment on the end of the line to a robust 3mm dyneema pig tail. That will be much stronger and you can use the pig tails for line length adjustment.