Ah, a Kent-Pattern, I'm not sure about this one was but quite a lot of these were made from a single flat section of mild steel, bent around a form to create the eye and a piece of high carbon steel sandwiched between the two cheeks of the axe and forge welded together. Not that that's relevant for what you're doing, just an interesting tidbit. I've re-handled quite a few axes and hammers, it's quite simple once you've got the knack of it.
First things first I would get a file out and knock off all that mushrooming off the head from what I assume is someone pounding it onto the handle from above? Once that is done it will be easier for the handle to project out from the top.
You should be able to re-use the existing handle although it will be roughly 1/2" to a full 1" shorter than it is now. What you need to do is put a sweeping taper onto it with a spokeshave or a knife, a similar sort of shape to half an hourglass in cross-section only less extreme, imagine the neck of the hourglass is the narrow at the centre of the axe head and where it splays out at the bottom for the sand is the same taper that is from the middle point of the axe eye to the bottom. From the narrow upwards you want it to be roughly parallel.
Once that is done, you want to put the head on and it should be a relatively snug fit, you then want to hold the handle in your hand with the top of the axe pointing towards the floor and give the handle a firm whack with a hammer at its end. The weight of the head wishes to stay in place in the air while the force of hitting the bottom of the handle drives it further into the eye of the head. This is a process of trial and error to get a good head seating, particularly if you haven't done it before, you eye up down the handle and make sure that the cutting edge is in-line with the handle and you have to take the head off and check where it's touching on the taper, if the head is a little skew-iff you can adjust it by taking material off the corresponding part of the taper to bring it back to be in-line. Generally, the eye of the head is quite dirty so it can leave quite helpful marks where it's touching so you can remove these spots to ensure that the bottom of the eye is making good contact all around the handle.
Once you're satisfied you've got a nice and snug fit on the bottom of the eye, run a saw cut down the top of the handle about 2/3rds the length of the eye and put the head back on the handle, driving it firmly onto the handle using the technique I mentioned earlier, you will actually hear the change in resonance when you've got it firmly on the handle as the handle will make quite a dead noise compared to the noise when the head is loose on the handle. Cut a new wedge that's a little longer than your saw cut, and tapers from nothing to a little wider than the two gaps on either side of the handle in the top of the eye combined and drive it in firmly with a hammer. You can put a secondary wedge in at 90-degrees or 45-degrees relative to the wooden wedge, these are usually steel and can be bought online for a couple of quid for a set of various sizes.
Once you've done all that you've got a proper tool
