This is really driving me insane.  I swear, it's like playing the piano with mittens on!

Okay, so I have this mesh.  I had a hole that wouldn't bridge, or fill or close using any automated tools, so I drew the proper shaped polygon.

Great, so I have a polygon the right shape in the right place, but how do I get it to attach to the mesh?

I'm in vertex editing.

It's really driving me bonkers, I can't find it in the manual or in an tuts.  Maybe if I knew what I even would call this made up tool.