There's a number of potential problems here, and they may not necessarily be at your end.
First of all, it should be clear that CMYK is not so much a generic "image mode" as it is a description of specific press conditions. If you produce final CMYK, you must have the right profile. Don't use Image > Mode, use Convert to Profile.
Given that, I would question anyone who doesn't accept Adobe RGB photographs for a magazine article, because it implies a process that isn't properly color managed and generally not up to date. A standard workflow today would use InDesign, where images are normally placed in their original RGB space whatever that is. Then the conversion to final output CMYK is done as the last step, when producing a press-ready PDF from InDesign.
It helps tremendously if you know that final CMYK, because then you can proof to that so your file survives the CMYK conversion with the least possible damage (we're talking gamut clipping here; not general color shifts). But it's not absolutely required, which is why this workflow is relaitively safe.
Third, CMYK jpeg is in itself risky, because many jpeg decoders (in some software) don't have proper CMYK support. The result can be weird colors. This could well be the problem here, because it sounds as if your client uses a non-standard workflow.
So what to do? I would send two files. One sRGB as reference, which is guaranteed to come through correctly. Then find a way to send TIFF for the CMYK version if they insist, or ask them to try opening it in different software. Of course you should have a calibrated and profiled display, but I agree that's probably not the immediate problem here. There's no particular reason a CMYK file should come back with twisted colors if it's opened and processed in properly color managed software.