If you can find the break in that coil, there's a neat method of stripping hair-fine wire without mechanically abrading it, called the 'aspirin trick'.
No lie, it really works. Put an aspirin tablet on something like a jar lid. Use a soldering iron (or gun) to heat the aspirin so it forms a puddle. Dip the end of the wire in it, and the hot acetacylic acid strips it clean. The fumes are gawdawful, so use plenty of ventilation.
