If I recall from my history reading, the Greeks and Romans had a medical tool(I think they called it a scapula after it's shape, but I am not sure) they used for getting barbed arrows out of people that looked like a long metal shoehorn(you stuck in in the hole along the shaft and used it to push and hold the flesh apart wide enough you could remove the arrow without causing further damage). Assuming that the Elves have such a tool (and it seems to me that assuming the Elves have a level of medical tech roughly equivalent to the Ancient Greeks and Romans does not seem much of a stretch.) someone with a very good hand could use such a thing to get the dagger shard out. One could re open the scar (remember Frodo's wound had closed up over the shard, so it'd have to be re-opened push in the scapula until it connected with the shard (a good hand could probably feel it hit) get under it to disloge it (sort of like how you can use a sewing needle to get a splinter out of your finger) then pull it back up on the scapula.
It also occurs to me that, given the circumstances, actually physically taking the the dagger tip out might not have been neccacary (I know Gandalf says it was removed, but just follow me here). Since the morgul blade dissolves in daylight, it might be possible that, if daylight was gotten to the tip, it would dissolve as well which might be considered safer (after all the tip was almost at Frodo's heart, so there was a real chance of Elrond killing Frodo in the process if he actually had to cut it out) If one is willing to let magic in in some form, maybe Elrond simply sent a beam of sunlight into the wound. HOW he would do that, I have no idea, but it occurs to me that, if Galdriel had the ability to trap the light of a star in some water held in a crystal container and have it stay there as an illumination source, Elrond might be able to take a ray of sunglight and direct it (maybe some sort of lens apparatus?)
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