On real pins, the whole playfield can lift up, revealing screws to remove what's called the apron. Then one can access the plunger assembly.

I fear this toy pinball was simply not made to be serviced...

Maybe your best bet would be to drill a hole in the old plunger pieces in order to firmly reattach the end with a screw through the middle, sparingly adding some glue when you tighten it in place. I can't tell if the plunger shaft is hollow; you might need to build it up with something in order to do that.