I say you cinder-block the sunken area whose inside dimension is 1 foot wider than the doorway. Coat and cap with cement. As for the sunken area, dig about 2-1/2 to 3' below your door level. Line the 'pit' with filter fabric (not landscape fabric) and fill with crushed stone to about 10" below the door sill. Wrap the excess fabric over top of the stone (so the stone is fully wrapped in the fabric.) Cut about 12" of a 4" PVC pipe and put a drain-screen on top, and stand it UP in the stone. (You can set it about 3" into the stone. DO cut a hole in the fabric of course!) Pour 4" of concrete where it will slightly slope towards your PVC in the middle. Do your steps if it's in the plan too. When it's harden, cut the PVC down to level and use concrete caulk around the outer perimiter of the PVC to seal it. Cut a small piece of filter fabric and make a decent size 'ball' out of it, and stuff it in the PVC hole. Then put the drain cap on. The ball will keep debris from 'contaminating' your drain. When you want to clean it out, just carefully pull it out of the PVC and shake it on the grass.
Now, I'd build the side-walls up high enough so water slopes away for a distance of 3'. The slope should be at least 1" over the 3' distance. The more the better. The same should go for the slop away from you home's foundation. If you can't avoid a negative slope away from your stairwell's side-walls, there's an asphalt-type board you can lay up against your outer side-walls to keep water (over time) from causing hydrolic pressure on your walls. This will ruin your well-walls! You can consider a relatively 'high' over-hang if you want to keep even more water out. Personally, I'd stay away from those storm covers. My best to ya and hope this helps.
PS: Filter fabric can be had from a quarry or a gravel yard. If you can't get it, ask them where you can.