Summary

8mm OD replacement pipe stem. Tapers from a 14mm base to a 8mm tip. Insert is 8mm OD, 10mm long. Stem is 110mm long.
Second File: Decreased IR by .5 mm for increased strength (first build broke off in pipe), Increased OR of insert by .25mm for tighter fit (first build was a bit loose).
Obviously you will have to adapt your stem to your pipe, or just use this as an idea for your own.