Your bike is a mid '60s English bike. Many of the parts on your bike are the same as on a Raleigh, Your bike was probably made by Raleigh Industries, Your rear rack is a Prestube Manor, used by Raleigh on 1965 and latter bikes. If your bike were mine, I would change the Sturmey Archer rear hub with a 40 hole AW hub and add a rear hand brake.
I agree that your butterfly handlebars need to be replaced with the traditional 3 speed handlebars. If you are going to change your rear hub, a set of new spokes, found on e-bay and made in China would be recommended at a cost of about $10. Both your TCW and the AW hubs measure the same diameter at 55cm. Therefore, you will need to remove one spoke and measure its' length, prior to ordering a new set of spokes. When you order the new spokes, ask the seller to ship you 40 spokes, otherwise they will only ship you 36 spokes!
Judging by the landscape in your photos, I presume you live in Orange, CA. I live about 60 miles north of you in Woodland Hills, CA. If you need any help, you can bring your bike north to my house and I could assist you with the upgrades!
Good luck,
Arnie