So you want to make a handle for a socket chisel without a lathe?
It isn't difficult, just tedious.
First off you need to rough out the shape. You can measure the diameter at the open end easily enough with a rule or calipers. Next you need to figure out how long and what taper. Get out your drill index, and use the butt ends of various drill bits to find one that bottoms out in the socket. This will give you the length and the small-end diameter of the taper.
Shave down the end of your handle blank to match the socket, then mark the length, and then the small diameter on the end. Shave down close to the marks. Once you get close, stick the handle into the socket and give it a slight twist. Don't spin it, just twist it enough to make the 'high' spots show up. I'll bet the inside of that socket is really filthy, so it will mark your taper quite nicely. Oh, and put a mark on the handle indicating which side goes "down" so that you always put the handle on the same way. The insides of those sockets are not necessarily perfect conical tapers!
use a scraper or file to keep taking the high spots off the taper until it gets stuck when you push it in when trying to mark it. You should then be close to a good fit. If you start to bottom out in the socket, trim the end of the taper a bit.
Rinse and repeat until it makes a solid friction fit. Drive it home.
Good luck. Take pictures, post them and let us know how it goes.
Wood Hoarder, Blade Sharpener, and Occasional Tool User