Mike
Most hitch pin plate holes are drilled all the way through. With a 5/32 nail set, just drive the old broken part out the back of the plate. If you use a piece of magnetic plastic strip, (as used on magnetic signs), to grab it on the backside, it'll save you fishing around for it. Yes, that stub will land somewhere to cause a buzzzzzz! DAMHIK.<G>
Insert a new pin and bend it over a bit. That will firm it up in the hole. Back off the existing string and hitch it up and bring it back up to pitch. Make sure you seat the string firmly on the plate/pin junction. The pin riding up on the hitch pin is usually why the pins break.
Best,
Joe