It has been that way for 27 years! The forced air was being fed into a huge pizza box space, uninsulated, with gaps that led the air to the basement, and probably outside. Probably cost me several thousand dollars in 27 years.
* I haven't shown the removal effort. There are other web sites, many, that show that.
Anyway, because of the peculiar set up, there was no easy way to buy ductwork. The duct was at right angles to the opening under the sink and fortunately almost aligned and not too ofset. So with a friendly help from a Master Plumber at Home Depot I constructed a new ductwork that fed the forced air to the register on the left. On the right hand side I added a support cross piece so that I could lay 2 pieces of the new base. For extra insurance, I added a bunch of insulation. The register is visible now and the orientation of the picture should be evident.
And now here is the finished work, with two pieces painted white. You can see a slight gap. I chose just to seat the pieces this way and if it doesn't work out I'll add some screws to anchor it. I left it that way in case I have to replace it again, only in another 30 years, I hope.
Note: How do you know you have this situation in your home? A fellow told me to see if the space under the sink was hot (in winter) or you can drive a small hole and see if there is any air coming out.