If its still working correctly, I'd just heat shrink it, which will encapsulate the jacket failure and create a protective strain relief. Do both mics for symmetry and to keep the other from failing similarly.
If its not working or intermittent (if intermittent, clean the plug first to make sure that's not causing the problem), you can cut/tin/solder and splice in a new cable. How difficult that will be depends on how tiny the wires are, how difficult they are to solder, and how close the cut is to the capsule. When securing the mic to do the soldering, if possible rather than clamping onto the capsule itself, clamp the wire between the cut and the capsule so that the clamp acts as a heat sink to protect the capsule a bit better during the solder work.
I'd do a test splice using some of the old cable you cut off the mic before going at it on the capsule.
Related-
I've a few miniature DPAs with similar assorted failures I intend to fix at some point. I'd prefer these were originally built with the microdot connector integrated into the back of the capsule, making the cable easily removable/replaceable rather than integrated. So may end up doing something like that on a few of them with failures close to the capsule. Something similar could be done using alternate connectors such as mini-plug or RCA. The DPAs use a coaxial cable and a few of these have a failure of the jacket and braided shield/ground yet the center conductor (which is somewhat difficult to solder on these) and the interior reinforcement fibers are in good shape so I may try to repair just the shield/ground using some small diameter braided-metallic Tech-flex I have on hand, then cover the entire cable with cloth-based Tech-flex to serve as replacement jacket.