Methylene chloride works great on many kinds of plastic for "welding" the pieces together and this appears to be a situation that if the plastic on your headphones is melted by the methylene chloride you can maybe plastic weld the pieces together. There may be a slight hazing/discoloration on the plastic parts after the welding process dries out(24-48 hours). The pieces must be stationary and not moved while welding it together and drying. If it is moved you will damage the joints and might not be able to compensate for the damage and the welding process will not be successful. It is important that the welding process is done in one try because if you try to weld it several times it makes a mess and sometimes the plastic becomes brittle at the joint if too much methylene chloride is applied over several days.
To use methylene chloride you use a very small pipette or a small phillips screwdriver to pickup a very small drop of methylene chloride and place the tip of the screwdriver over the cracks and the capillary action will suck up the methylene chloride from the tip of the screwdriver or pipette. Don't bother using the alternative method of little squirt bottle with the small metal pipe for application because it will not be effective in your case. It is difficult to control the quantity and makes a mess using the small bottle with it's little metal pipe. In your case just use the screwdriver method to apply the methylene chloride.
Before using methylene chloride one should always test it on a small section of the plastic in a spot you won't see after it dries. If the methylene chloride melts the plastic then you have the kind of plastic on hand that can be welded together with methylene chloride.
It is available in small quantities. You will only need maybe at most 5-10 tiny drops of the methylene chloride to get the meting process in action and join the plastic breakage together. It can be found at industrial plastics shops.
Methylene chloride is very thin like vodka. Thinner than water and so it requires a steady hand to apply or it will fall off the tip of the screwdriver or pipette.
Like I said earlier test the methylene chloride on a hidden section of the plastic. Let it sit for 2 days to thoroughly dry and then see the result.
Don't apply too much. Like I said just a few drops works best if the break/joint is very close together it works very well. If the break/joint is too far apart the methylene chloride will not stay in the joint and will simply fall out. So make sure the joint to be welded is tight and as close together as possible. It takes a minute to a couple of minutes for the methylene chloride to melt the plastic so be patient.
Use it in ventilated conditions because I think it might be bad for you. Keep the lid on tight or it will evaporate right out of a loose lid. It evaporates fast if on the surface but when it gets in the plastic like I said it needs a couple of days to dry properly.
Be sure to test it first before attempting the full weld of the break/joint.
https://en.wikipedia.org/wiki/Dichloromethane
https://www.youtube.com/results?search_query=methylene+chloride
It's also used to make decaffeinated coffee.
