This is a tough one given you have tried everything that I can think of except a a pick. The type dentist use or even those plastic pieces with a small piece of floss. When I first say them I thought No Way! But the actually are quite sturdy and effective with mouth wash for just the problem you describe. I know I have a bridge that always seems to get the difficult piece of everything and I have to do this quite often and the dental piece with Listerine always seems to work for me. That of a few good stiff shots!
Tooth pick or tooth floss:)
Look come down go and see a doctor this happened to my cousin she couldnt even talk she went to see a profesional doctor because it was deep in the gum trust me on this one!now my cousin could talk!
Can't add much more to this other than if you do pull your tooth out you better be quick about it because stopping half way is not an option.... Trust me on this lol