Each tread will need to be slightly different in size. Measure each tread and cut it to size then install it using 2-1/4" trim screws.
Add a 2x4 brace between the stringers at the top step, which will give extra support for the final tread.
Staple down 15-lb. roofing felt to the area. Install the nosing border (image 1) that's cut at a 45-degree angle by pre-drilling the wood, and attach with 2-1/4" trim screws. Since the nosing has a groove facing out, insert a spline (image 2) in order to create a tongue for the wood flooring to connect to. Use a rubber mallet and protective block to tap the hardwood planks into position, and then secure it in place using a pneumatic floor nailer. Stagger the planks as you work across the landing. The wall may prevent the use of the floor nailer on the last few rows, so use a regular nail gun with finishing nails in the edge of the boards. Rip cut floor boards for the last row and nail in place. Finish the landing's new hardwood floor by filling the holes with wood putty.
Pre-drill holes in the post and holes for toggle bolts in the wall. Insert screws for toggle bolts and attach toggles on the other side. Insert the toggles into the wall and tighten the screws
Notch the newel post's bottom to fit snugly around the step and secure it with lag bolts. Attach the bottom newel post goes in the same way.
Pre-drill the landing and treads for the balusters, using a special stair bit that drills the hole and then creates a thread to accept the baluster bolts. To prepare the balusters, drill out the center and insert the baluster bolts in the bottom. Screw the balusters into the floor.
Attach the rail fasteners into the newel posts. Loosen the newel post and set the railing in place first by guiding it onto the rail fastener, and then by inserting the balusters into the holes in the railing. The angled portion of the railing may require removing the bottom newel post. Lining up all the balusters and holding the railing in place goes much easier with two people. Re-attach the newel post and tighten up the rail fastener.
To complete the project, install the new base moldings all around the stairs and landing.