I used the frame I had, since it's strong and the wheels are still rolling well. The inner width is 29 1/2" so I planned to make each of the three laundry bags 10" wide, and about 14" deep. The height of my bags was about 24 1/2".
I can't tell you how great it felt to throw that old ripped sorter into the trash. Note how the ties are all broken and the separators are torn down the sides.
With the remaining 14" strip, cut off a selvedge edge and then cut a piece the width you've just calculated above. For example, I had 43" of usable fabric on my first fabric. Using the equation 50-43, I needed a 7" piece from the second strip. My second and third bags started with 41". I needed 9" pieces from the second strips on those.
Next, take the remaining second strip and cut a 25" wide piece, (by 14" long,) then cut that lengthwise in half to make two, 25" by 7" strips. The remaining fabric from that second strip will make your ties. If possible, cut 4 strips, 2" by 14" but if not, you can cheat and make them skinnier. No less than 1 1/2" though, I'd say. If you cut the other pieces sparingly, you should be fine.
Sew the 14" long pieces together along one of the 14" edges using 1/2" SA. Similarly, sew the 7" pieces together along one 7" edge, making sure that you keep directional prints the same direction. Press the seam open. While you have the iron hot, press the ties in half lengthwise, then each side into the center. Then press in half lenthwise as if you are making double fold bias tape.
At this point, if you want to use interfacing to stiffen the fabric components of your bag, apply fusible interfacing. I made the base of my first bag and then wished I'd used some interfacing so I added it to subsequent pieces. Truthfully, I'm not sure it matters. If you want to interface, use strips 13 1/2" long by WOF for the bases, and 6 1/2" by WOF for the upper rings. I had to piece interfacing onto the fabric but am too lazy to piece the interface before fusing it. It really doesn't matter. Complete the rings by sewing the ends together, right sides together, of both the 14" and 7" strips. Press seams open.
*By "top" and "bottom," I am being careful here to make sure my print fabric ends up right side up. That matters only if you fabric is directional. Check out this photo: the narrower strip is going to be along the top edge of the laundry bag, and the wider strip is going to be at the base.
Sew along long and short edges of each of the ties, a scant 1/8" from the edges.
Lay out the mesh bag, inside out, and cut the bottom and top off. You need a 27 1/2" long tube, the full width of the bag; if your mesh bag has rounded corners at the bottom, try to use above that section. If you can't get 27 1/2" of full width, just slit up the curved seam so that the bottom opens all the way.
Now create the boxed bottom of the laundry sorter bag by aligning the side seam with the bottom seam that you've just created. Using a ruler, find the point along the seam where the width of side will be 10", and draw a line across the corner, perpendicular to the side seam. Sew across this line, backstitching at the ends, for strength. Do this to both bottom corners.
Before turning the bag right side out, take the opportunity to put a few pins in the bag to attach the fabric to the mesh at the pressed top of the fabric. Turn right side out and put more pins in to affix the mesh to the fabric. Topstitch the upper edge of the base fabric to the mesh bag on the right side.
Now, you can either cut off the base triangles leaving a 1/2" seam allowance, (Serge or zigzag this to finish,) or pin that triangle down into the base of the bag and stitch it down to offer a little extra bulk and stability. I sewed them down into the base.
With the bag now right side out, insert the upper fabric ring, right side out, so the raw edges are aligned. This feels weird because you are pinning the right side of the fabric to the wrong side of the mesh. Reassure yourself by checking; if you sew a 1/2" inch seam along this raw edge, will the fabric later flip around and fold down to make the ring with only right sides showing?
Now, to attach the ties, I'd try the laundry bag on the rack by holding it up there and estimating where you want the corners to tie, and how well everything is going to fit together. Pin those babies to where you think they belong, and sew them on securely to the corners, just right!!