Since the wood flooring runs parallel to the doorway, it shouldn't be too hard to take off the sheet metal, patch the subfloor, extend the wood floor with a bullnose/stair nose at the edge and finish it to match. There might be a slight mismatch to your existing floor but it would look a million times better than now and 1000 times better than having carpet go up the step and extend through the doorway.
You might even be able to find prefinished flooring and a nosing that matches you existing wood closely enough that you could avoid sanding and finishing.
Having carpet there might cause a trip hazard since people might not be able to discern where the step is. Also, the carpet at the edge of the nose is likely to wear faster than the rest of the carpet.

Well, that's different!

I agree with JFC that carpeting it would make me trip. :) Filling it in like he suggests would be the best option. Treat it as a step, but not a carpeted one.

Is the old floor register still connected? Does that metal thing get toasty when the furnace is on? If it is, it might be worth it to cap it off up the line a ways. Not to imply that the previous owner would cut a corner!