I've read your reply about the part you found in Home Depot not being the right size. My solution is that you buy this anyway plus a selection of different sized nuts and bolts. Use this selection of nuts to find out which one fits the thread size on your stopper rod and which one fits Home Depot knob. You will then know what sizes the two different threads are.
You can use this information to find a "thread adaptor" online which has a female end that matches your rod and a male end that fits the Home Depot stopper.
When you have found the right thead adaptor and had it delivered, it's simply a case of screwing the thread adaptor onto the rod and then the knob on to that.
Whilst you are waiting for the thread adaptor arrive, you can use the nut that fits the rod as a temporary knob, albeit very small!
This fix will only cost a few bucks and hope it is the solution you need.