Oh, that's a really nasty place for the thing to break! It makes the problem difficult.
Bending aluminium alloy has been mentioned. It's worth noting that many aluminium alloys can be annealed by heating. To do this, smear ordinary soap onto the surface, then heat (blowtorch or even hot-air gun) until the soap chars. The alloy will then be softened and can be bent, later, when cold, with much less risk of cracking. Beware, though: the heat treatment of aluminium alloys is complex, and without knowing the alloy's composition, the best one can do at home is the crude method above, which will probably leave the part softer, thus more easily bent, than it was originally. Some alloys will re-harden over a few days.
Regarding adhesives: a butt joint will be weak, because of the small cross-section. There's not much thickness for internal reinforcement. If I were going that route, I'd consider cutting a long slot in each piece, rather than drilling a hole, then the reinforcement can be the full width of the key rod. If there's enough room to stick doublers onto each side, across the crack, adhesive will probably work, especially with a rubber-toughened cyanoacrylate, like Loctite 480. The doublers should be as deep as possible, but can be thin material. Meticulous de-greasing is always advised for adhesive fixing. I wouldn't worry too much about dissimilar metal corrosion, if there's no electrolyte around. This means keeping moisture away, and a coat of waterproof paint, varnish, etc. will do the job. In any case, the alloy key rod is in contact with the brass piece - a prime candidate for galvanic corrosion!
Having said all that, I note that, after repair, the brass piece will have to be rivetted to the alloy key rod. This is risky for brittle cyanoacrylates, hence the rubber-toughened one advised. That might even allow rivets to be dispensed with.
I think that removal of the brass piece, followed by a favour from a friendly - and skilled! - TIG welder is the way to go. However, the welding will anneal the alloy around the weld area, so it will most likely be softer than original. Spectacle frame repairers often have laser welders these days, and will often do jobs other than spectacles for reasonable prices - and an offer of beer might help.... I don't know, however, whether they will be able to attack aluminium. If they can, it's great, because the heat-affected zone is so small, so little strength will be lost.
