You can use a regular sewing machine with zig zag stitch and denim needle. the zig zag width is adjustable on most sewing machines so you'll have to play with the adjustment, the idea is to stitch the two lines together under the sleeving. So adjust the width to accomplish this while keeping the threads off the outer edge of the sleeving as this can be chafed when making a larks head.
You need to use thread that is UV and mildew, water resistant and reasonable strong, I use sail repair thread. Have made dozens of line sets none have failed becasue of my sleeving or stitching. Also use denim needle,
In a larks head knot the thread is virtually unloaded so this is not a high stakes game if you're stitching isn't perfect. Or if the thread isn't bombproff I ride my shit till my lines break or till I start losing strands and NONE have failed at my sleeving
The claw on most sewing machines will be too narrow or wide to feed the sleeved kite line through consistently. You will have to make a cardboard template with a length cut out of it equal to the length you want to sew(the cut out is so you don't sew the line to the template)
Then you tape the sleeved kiteline to the template. The claw then pushes on the template to feed the assembly through the sewing machine. For the template think light switch cover only cardboard with a longer cut out.
the stitching is straight forward back stitch first centermeters or two, run it the length of the line and back stich the last centemeter takes some practice so try it on a piece of broken line and plan on discarding 1 sleeve, (you can get more at www.fixmykite.com
) Use a thread ripper rip the threads out, try again till you can make it look pro consistently. if you can already sew, this should be no problem,
Really only tricky part is figuring out you need to use the template.....otherwise you're just running it over with a sewing machine