Good Morning:
I see that you have several replies and you were successful in removing the broken plug; however, if in the future you do not succeed in removing the broken fitting plug, there may be a simpler way to correct this issue.
Not all, but many HPLC columns these days have removable end fittings. For example, Supelco HPLC columns have removable end fittings, and the fittings used on 3-4.6mm ID columns are interchangeable. Therefore, if you are working with columns that have removable end fittings, when a column dies, remove the end fittings, wash them thoroughly, and place them in a brand-labeled box or plastic bag for issues like this one. Then all you need to do is replace the end fitting, no digging out the broken plug. Before removing an end fitting from an HPLC column, determine if the frits are flat or if the frits are capped. If the column has flat frits, nothing stabilizing it so that it does not move when the fitting is removed disturbing the silica gel, I would avoid removing the end fitting if at all possible. However, it the column has a "frit cap", such as Supelco HPLC columns, the frit will not move when the end fitting is removed, resulting in no disturbance of the silica gel.
Regards,
Linda