What's the best adhesive to repair a snapped plastic indicator stalk?

Drilling and inserting a headless screw or even a bit of welding rod can work.

No adhesive or resin out there is going to solve this problem by itself. Plastic is non-porous, and a snapped piece of plastic won't have much surface area to bond, so you won't get much strength even with a high-strength 2 part epoxy resin. Even with epoxy you may be able to get it to stay in place but it will simply snap when you try to use the turn signal.

You need to add some strength using some sort of rigid material inside or outside of the stalk. If you had enough of the stalk sticking out I'd recommend doing it outside, using piece of drinking straw or something similar to act as a splint. You'd fill the straw with epoxy and then put it on the parts of the stalk. Epoxy would squirt out and it could get messy, but it would be effective.

As you say there's no stalk sitting out you have a slightly more challenging fix as the only way to give rigidity is to put material on the hollow inside of the stalk. You'll want to find a rigid piece of metal pin (think a nail with no head, maybe a screw) that just barely fits inside, then epoxy both inside to fix that metal in place and at the joint as well. Use a 5 minute epoxy and work quickly, just don't go crazy and pour lots of epoxy in your steering column, just just enough.

Also, when using epoxy make sure you have good ventilation and use gloves, epoxy is awesome stuff, but it's not good for you!