Sew the two strips together on the diagonal -- just like when joining all the strips together for the binding -- to get that diagonal seam that will lays flat.
Trim off the excess fabric to have just a 1/4 inch seam allowance.
And the binding finished off the exact right length and lays nice and flat, with a perfect diagonal seam -- just like the rest of them. You can't even tell which seam it is from the rest of the seams.
Now -- that just couldn't be simpler. I am so happy with this method. Probably everybody - or should I say - every Quilter - already knows this method and I am just the last to discover it. But in my various books - etc I have never seen this method, and it is by far the easiest way to join the ends I have ever done.