Floor Support Repair
I had a similar situation, only it involved removing some load bearing walls.  The wall framed what originally was the furnace (boiler) room in the house.  Inside that room was a ~3'x6' entry into the attic.  So the wall on two sides of the room provided the support for the ceiling joists, similar to the way your floor joists rested on the old wall for the stairway.

When I removed the furnace room and it's walls, I had to provide a new support beam for the ceiling joists, which are also the floor joists for the attic above, to replace the load carrying walls that were removed.

I cut back the existing joists enough so I could get a doubled up 2x12 glulam up there.  Then I fastened the existing, cut back joists to the glulam beam with joist hangars.  I also had to provide required posts back in the two wall that the beam rested on, which carries the load down to the foundation.  Note that I had to do this two times because I did not want to move the access to the attic, and had to provide the beams on both sides of the attic opening.  Here are some pictures:

First shows the existing joists cut back to provide room for the beam.



Second shows one of the two beam in place and the joists attached to it-



Here's a view of both beams.  Note the double or tripled-up 2x4 posts added in the existing wall to carry the load from the beams.



And yes, the flowered contact paper is still on the hatch covering to the attic.  Wife thinks its whimsical.

I can put together an overall sketch if people think that will help

EDIT 1 - Added sketch of joist layout