Okey, the first batch I made, I used my standmixer, since I did not have a stickblender. I made a 100% olive oil soap, and I had to mix it on the highest setting for 3 hours! Yes, 3 hours. I maybe could have stopped before, but I needed to make sure I had trace. And I tried a handmixer once. The soap did not come to trace, it was separated in oil and lye, just like in your picture, so I added more lye (because then I thought the lye would be neutralized) and it came to trace (but was lye-heavy and had to be thrown away). So with a handmixer, you will have to mix it for ages at no less than maximum speed, but it will come to trace eventually. I suspect you have mixed for a long time, but not nearly as long as it requires with such tools. Maybe you should use a smaller bowl for safety, so that it doesn't splatter everywhere. A jug or something else that is narrow and deep instead of wide and shallow. It might be a more efficient mixing in a jug as well.
You can try placing the jug in a bowl of hot water while you mix, to heat up the soap a bit. That should in theory reduce mixing time, since heat speeds up saponification.
Maybe adding a tablespoon or two of vodka or other strong alcohol could help speed it up? I'm a beginner myself, so this advice you have to get verified by someone else, since it might do something very unwanted. So it is just a suggestion.
I have read that you can add some ground up soap to help with emulsification. So if you have a bar of soap, just ground up some of it and add it, and mix at full speed.
Or you could run out to a store that sells stickblenders, if you have such stores near by. A stickblender is the only thing that really works fast and efficient. But a handmixer or standmixer will make soap too, it just takes a very long time. But it is definately possible. Happy soaping!