A crude fix might work via a small piece of easily removable tape (masking or painters) that you could apply as a "cast" to hold the wire in place while the wire is in contact. Put the headset on, bend to make contact (left speaker works) then tape. Long term success not guaranteed.
May take a couple of tries to succeed so that is why you would use removable tape.
More of an interim/temporary repair but a family member's college speakers had a couple of such "repairs". One speaker even had small wire nuts compliments of a EE major. Still in use.
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Can you tell if the left speaker will open up so you can see the connections inside. Maybe small screws, press tabs - perhaps under padding or some label.
Having just a couple of centimeters of wire to work with is pretty tight - no room for error as the insulation will need to be stripped above the break.
If the left speaker can be opened then you might be able to desolder or otherwise disconnect the wire ends, cut off that "last couple of centimeters", then strip and reattach the remaining wire end. (Provided that the wire is still long enough to do so as every extra centimeter extra costs money. If not then you would need a matching wire to splice in and thus regain the length. Doubling the work.)
And there will actually be two conductors to splice. The braided/stranded wire is very fine and it is very easy to lose pieces of it while removing the insulation. Especially with a small wire stripper. A razor blade or X-acto knife might work better. to remove the insulation. The idea being not to cut or knick the thin wires making up the strand. Rejoin the connections with solder or heat shrink tubing. Either way can be problematic for a variety of reasons.
Here is a link with images:
https/www.google.com/search?q=how+to+splice+thin+stranded+wire&biw=1282&bih=691&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwjOlvGK8efOAhWKvBoKHXF6DQoQsAQILw
Google "splicing stranded wires" in order to get a visual idea of what is involved. Way too much to effectively cover in just words. The idea is for you to get a real sense of what is necessary for such a repair.
Hopefully you will have a family member or friend with some electronic skills via work or hobby. Go that route first if you can. If help is not possible then you can try a splice yourself. Just keep in mind that a mistake may be the end of the headset.