I did a rewrap on this 1840 cavalry officer saber, which would probably be similar to your 1850 foot.
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Here is a long explanation that will likely be boring to everyone except those actually doing the task. My sword still had the original thin single strands of flanking wire but was missing the twisted cental wire. I ordered the tarnished version from Tom Nardi, and he does send instructions with the wire. Not sure where your wire is broken and how much is missing, because if the portion that goes under the pommel cap is still there and you can match that wire, it may be easier to leave that portion in place. All my wire was missing. I started at the front end near the hand guard. There should be a little hole where the original wire was inserted. I enlarged that hole just a bit with a thin nail, inserted a tiny drop of epoxy, then the end of the new wire, and then pushed in a tiny wedge of wood like a toothpick tip to hold everything in place. (Carefully trim the wooden wedge flush and dab with some brown stain to make it look aged when the wire wrapping is complete.) Wrap the wire in the original grooves towards the pommel cap. Now here's where Tom's directions tell you how to cheat if you need to get wire under the pommel cap and don't have the ability or want the hassle of taking the sword apart. When you get to the groove where the wire first passes under the pommel cap, cut the wire you've been wrapping just a bit longer than where it meets the pommel cap. I used a nail again to slightly widen the space where the wire enters, injected a drop of epoxy, and then pushed the wire into the opening and under the cap. For any remaining grooves you do the same thing- cut a piece of wire just slightly longer than the groove' then prepare the openings, inject the epoxy, and push first one end and then the other under the cap. Err on the side of cutting your wire a bit too long and push as much in as you can. You can always trim, but can never make it longer and you don't want to run out of wire. So my rewrap was really done with 3 pieces of wire, a long one from hand guard until meeting the pommel cap, and two short ones in individual grooves.