The drill is not designed for the side loads that a router bit imparts; but hell, if it is just for this once to get the job done then I'd stick a dog turd in the drill if that would do the job.
As long as it's a plunge bit, you tightly clamp the workpiece, and as long as you don't care about appearance it will definitely work. If you have a drill press, that would work better, obviously. It has to be a plunge cutting bit.
I don't think 850rpm is enough to run a router bit.
It'd be a great excuse to get a Dremel tool. I've recessed lots of things with the dremel tool and a square-flute cutting bit. They even make a depth fixture that screws onto the front of it. Hand-held, it's exactly what you're looking for.