There is part of the confusion. The old one broke or the PO cut it off. A leach line is nice, but not essential. If you are feeling rich, have a sailmaker install one. He'll probably try to sell you a new sail instead.indeed - it seems there's a cord of some sort inside the sail along the leech - but there's no loose line anywhere on this thing. having no line is like having no car and being asked to demonstrate parralel parking.
That thing you're describing is the head of the sail. Typically leach lines are wholly encased in the leach except where they come out to a cleat. Take a good look at the leach, right below the reef points. Look for a row or two of horizontal stitching, that is probably where the leach line was attached.There's a large metal plate at the top corner (yes i know thats not the right name for it, i know the clew name at least!) with a number of holes in it which i PRESUME is where such a line goes but maybe not? I mean, one of them has to be for the halyard itself ... logically.
Luffing is what happens at the luff (front of the sail). The traveller, vang, and mainsheet can affect leach tension, but if the sail is older and stretched out, that may not be possible. Then the leach line can take out the flutter. Get the boat in the water and start sailing it. This will all make more sense when you actually sailing.If its luffing wouldn'y you eliminate the looseness of the leech by pulling a bit of tension on the main sheet... maybe a little in the vang? and if that doesn't work, adjusting the traveler so the main is pulling down more on the boom, instead of holding it in? If you pull a line on the leech thighter i would think that would introduce more slop to the tension of that edge of the sail.