It would need a precise retrograde burn at a precise orbital location. This means it will be slower than surface based nukes and have a highly variable time to target. At a low orbit this could be 90 minutes, at geostationary orbit it could be almost a day, but it will always vary based on the location of the target and the weapons orbital position when the order is issued.

I think they'd just get very strategic about the direction they launch it from the satellite. They mentioned 15 minutes in the article but that would require multiple satellites in the sky. The idea of putting rocketry and guidance systems omboard seems plausible too.

You can't deorbit something just by "dropping" it from a satellite. If you want to target a single "rod" they would need independent onboard guidance and propulsion.

A better orbit is retrograde solar, just inside or outside the Earth's orbit. With 183 of them in orbit, one would pass by the Earth every day. With a week or so advance notice, aiming would be easy.