Find a Handle
You will need to go to a hardware store to find a replacement handle and its much easier to do this if you bring the axe with you to match size and fit. You will also need a few wood handle wedges and twice as many steel handle wedges.
Remove the Handle
Saw off the broken handle close to the head of the axe and soak the head overnight in a bucket of water and then let it dry for a complete day. The water will swell the wood but when it dries it will be looser than before.
Clamp the Axe Head
If there are wedges at the top of the handle knock them out with a hammer and chisel. Next open the vice and put the head back on upside down. Now begin tapping the handle out with the largest diameter steel rod. Work on multiple spots for the best results and use a smaller rod if corner areas get stuck.
Check if then new Handle Fits
Keep the axe upside down to check if the new handle will fit or not. If it goes in too easily its the wrong handle. It should be a little too big for the axe to fit correctly. The handle should have a saw kerf cut across the longest axis. The wooden wedge you bought should be a little shorter than this slot. The sides of the axe socket are not parallel and towards the top the opening widens. Use the rasp to carefully trip the handle down until it matches the axe head socket. Now put the handle through the head.
If there is excess wood above the head saw it off as close to the steel as possible. Next put the steel wedge through the axe handle from above the head. This will expand the handle to fit the head.