I need to figure out how I'm going to plumb it, then go back and buy pipe. Also got some rope and some gloves. Pulled out the old sump pump. Verified that it was the pump, and not the switch, that was broken. The just-a-pump pump I bought earlier is now down in the sump and has fully drained it (at least as far as it's capable). But it's refilling fast. I was going to try to connect the switch from the old sump pump to it, but I can't get it threaded onto the rope, and have no other good mechanisms for affixing them. Woulda been a good stopgap to avoid going back to home depot again, and trying to work out a way to get the water out.
It appears that the old sump pump has the same sort of outlet as the new one: a 1.5" pipe. The installer of the old pump was bright enough to put an adapter on it in order to connect it to a smaller hose (.5"? .75"? thereabouts) which runs across the wall to the sink. I hate to even consider continuing to use their stupidity, but the only other option at the moment is to run a 1.5" pipe out the window. Since running it across the sidewalk is out of the question (the hose on the current pump goes there, but it's easy to step over), I'd just be pumping it right back into the house, effectively.
I'm going to read the manual for the new sump pump and see if I get any bright ideas.