Step # 1 is to check and see if the broken piece is stuck in the carburetor.----If it is you need to get it out.
open up the top dime size plug and you may see the tip of the needle in the side hole sticking out in the drip chamber push it back out
I it just flush with the wall
Are you saying the tip of the needle is in there but not sticking out into the cavity, so you can't push it back out? I've saved carburetors in such conditions by drilling a hole into the cavity, opposite where the needle piece is lodged. Then you can get a straight shot at the piece with a suitable punch and knock it out. Needless to say, now the hole you drilled has to be sealed. Epoxy.
You're sure it's in that hole? Try a flashlight in front and see if the light shines through up by the removed compression fitting. No need to show your mettle here -- object is to repair it in the simplest/least damaging way possible.