Mmmm. Been looking at the bobbin. Can the bobbin slide of the core? Why buy new wire if the old wire is still good? Is that for a wipika coil?
HT coils I have found faulty was usually the thin wire burning off near the "outlet" tag. Sometimes a sharp edge somewhere on the tab causes the coil to arc there, and eventually melt through the secondary wire. I have fixed many a transformer this way:
Remove the outer layer of insulation, get to the first layer of the secondary, scrape and clean the wire end nearest the tag, check resistance over primary. If you get good resistance then you've found your trouble, unwind a turn or two, resolder the secondary to the tab, make sure you leave no sharp edges, and bob's your uncle. Of course I have no idea where the actual tab is for the HT outlet on these coils!
Usually its as far from the primary as possible on the outside of the coil. Of course if this is not your problem, you will have to unwind. Unwind a layer at a time or until the broken wire comes off by itself, chances are the damage is in the outer layer. If the coil have burned (shorted) the resistance will be much less than usual, in this case the whole coil needs replacing.
The insulation layer is not as much insulation as to prevent the coils of thinner wire to slide into the thicker wire. Almost anything can be used for insulation. Wax paper/tape comes readily to mind. The windings need to be in the correct direction else the voltage me be out of phase, I think its the same direction as primary, but can't recall offhand and I am too lazy to google, search "autotransformer winding" or something or check your now opened up transformer.
If the bobbin can slide off, jiggle a steel shaft in there. Clamp it in a vice and use a drill or Dremel with a shaft to unroll the old coils unto it. Once done, measure the resistance over the length of wire. If the resistance is highish, dunno, I've heard 4000ohm mentioned for these coils, then your old wire is ok for re-use. I may be wrong in removing the wire, because it may be glued in, but that is seldom the case except for the outer layer or two, because of sticky glue in which case you can soak it off. Do not use anything that dissolve varnish though
. Usually coils that gets too hot have the winding's insulation melt a slight bit and then stick to each other, thus you will have plenty of breakages, so just cut it off.
For secondary wire, an old car coil is usually easy to open. (Scrap yards have plenty, of course they will sell it as "good" condition.) Just bend the lip of the can open You will find oil and plenty of wire inside. It may be too thick though, but it should be correct. The windings on a car coil is very easy to unroll, since it sits in slippery oil! If you are clever you can unroll straight from the car coil unto the seagull core. Precise winding should not be a problem, as long as you don't bunch up the wire too much. If you can add more windings than original. You will have a higher voltage, which should give you a better spark! I don't think you will be able to "overwind" the coil so that HT breakdown will occur. The voltage will in any case vary much by how strong your flywheel magnet is and how close it comes to the coil. "Charging" the magnet with a drill is a typical application of spark strength to magnet power.
If you can figure out the primary voltage, it will be easy to work out how many secondary turns is needed. For a typical 12V coil, to get a 12000V spark you need 1000 turns of secondary for every turn of primary. If you want to be daft and want to reduce the number of secondary turns, increase primary voltage. Perhaps a 12V battery in series with the points? Heh heh heh!
Or else a stronger magnet, or remove a coil or two from the primary.
