IMHO it's past due for a new nut, since repairing a broken nut that broke under a string is going to be "challenging" at best. And if it's plastic, old plastic gets brittle and may break elsewhere. But given you are trying to preserve as much "all-original" as possible:
If I wanted to attempt a repair of the old one, I'd pop the rest of it out, clamp the parts together, shave or sand the bottom slightly, mix up about a gram of epoxy, un-clamp the parts, dab a tiny drop on the broken surface, re-clamp the parts together, and add a carbon fiber tow strand to the bottom surface with more of the epoxy, then let it cure, un-clamp, clean up, and re-install. Be sure to separate the nut from the clamps with something like packing tape that epoxy won't stick to.
The added carbon fiber and epoxy (both quite thin if done right) are why the bottom would be slightly shaved. The carbon adds tensile strength to resist the string's natural tendency to break it again at the small-area glue joint under the string. Doing it this way means you haven't messed up the neck by trying to glue the broken bits into the nut-slot so firmly they won't come apart again, and if the nut is replaced in the future it should come out as easily as any normally "or or two dots of glue" installed nut.
If the old nut falls apart when removing it, then I'm back to "brittle, time to replace it." At least assuming it's supposed to be playable, not just a no-longer-musical artifact.