Is there any way to salvage a broken needle (discrete form-factor, hard to replace)?
Most sewing needles are steel of some sort (possibly stainless), which doesn't solder with ordinary electronics or plumbing methods. If you have the kit and experience, it might be possible to braze the steel, but only if the break is in the right place. You need somewhere to build up material, so you can't treat it like superglue. You could of course file away material before and after.

If it's a fairly big needle and I was desperate, I'd make a jig to hold it (clamping a cork to the machine should do) and use a very little bit of slow epoxy between the broken faces. Look for one that tells you not to handle the part for several hours, and pay attention to the "full strength is reached in" instruction. You may be able to build up the join on the outside afterwards.  This might be enough to finish the job, but it's unlikely to be a long term fix.