Nothing to grab eh! Oh well here's an idea. There is a tool called a stud extractor used typically on car engines. What happens is a stud has sheared, you drill a hole in it of the correct size then screw in the extractor. It is reverse threaded so once screwed in you simply undo in normal rotation and out it comes. Bit over-gunned for your problem? No, the extractor typically comes as a set, maybe 4-10 different sizes in the box. The skiniest ones would be ok for you. You already have the hole through the spigot. All you got to do is be careful not to screw in to hard initially. Once firm gently rotate and it should pop out. Extractor sets are cheap, look for a good tool shop.
Plan B) Bit more delicate this one...presumably the broken section (long bit) is now scrap. Ok, shave the eyes off and you now have a mandrill type thing which can be inserted into the broken piece inside-a tapered fit. Try it in the hole and mark it's depth where it starts to become a firm fit. Go get Araldite Rapid adhesive. Apply a
thin coat and re-insert. Wait 1 hour and yank it out! Be careful to only apply Araldite thin because before setting it can run!!!
Good luck
Chris.