Pto shafts are all different, different ends.
I would priced in the cost of a new shaft and left the pto with your old mower, both of you would be happy, now neither of you has anything that works.
You can get a new 1/2 yoke and remove the 2 endcaps on a joint - little fussy not to loose any needle bearings as you swap, or you can buy a new shaft with the proper ends.
An adaptor is not a normal way to do this, you will change the geomitry of the driveline. Your shaft will need to be shortened, and the adaptor would stick out, creating stress from the unsupported length and steeper angles with the shorter shaft. I don't think you will have good results with this path.
Pto shafts are fitted to, and come with an implement. There are many different end styles, including built-in ratchet or slip clutches. Something to consider in the future, always get the original pto shaft or discount the purchase price a fair amount.
--->Paul