I suggest that the first approach is to remove the fragments of wood from both pieces. Once that is accomplished you might be able to determine a near-standard diameter that will accept a replacement wooden dowel. Measure the depth of both pieces, add them together and cut that length of dowel.
If you don't have a standard diameter, buy the next larger size and sand it as closely as possible. A polyurethane glue such as Gorilla Glue or equivalent will swell up to fill any small irregularities caused by sanding.
You may also be able to sand a slight taper to one end of the dowel and use a hammer to tap it into the metal section, shaving off the excess, then sanding away the portion that extends outside of the metal section to match diameters.
If you want extra-super strength for the joint, there is a product known as Fiberfix, available at many handyman type locations, including Ace Hardware (US). It appears to be an air-curing type fiberglass tape and resin, activated after a period of time removed from the air-tight packaging. The video adverts are encouraging, but I don't believe everything I see on the internet.