Without a scale, both of these look about the same to me. I couldn't make anything out of that except that it looks about like what I'd expect as the voltage at the top of the LED stack with the capacitor in place (__if__ the amplitude were small... on the order of tens of millivolts.)
This is instead the voltage across a 4 ohm resistor (which should also look like the voltage at the top of the LED stack), but the ripple height should be low -- on the order of millivolts, too.
Okay... that's good.
Okay... that's bad. This suggests 1V/4ohm = 250mA variation. Which sounds very bad to me.
Okay. That will take some serious time to charge up, if it is working okay. (1/4 second or more.) I hope it has a high enough voltage spec, too! If not, bad news there.
Sounds like a changing situation. Parts are getting hammered.
Perhaps the better thing to do right now is to replace your LEDs with a resistor! And play it safer until the darned thing is working right.
With 6 LEDs running on say 30mA and call it 21V, you need 21V/30mA or
700 ohms. That's going to burn off over 1/2 watt, too! So make it a 2 watt resistor. (Don't use an 1/8th watt, unless you use a LOT of them paralleled up.) 700 ohms is hard to find, so use something in that area or parallel a few to get close to it. Just keep in mind that power figure! It's a lot.....
Maybe someone can do somewhat better than me, looking at your pictures. There is a sharp bottom in both pictures. But I don't know if that is located at zero volts, or not. However, I would guess it isn't at zero volts because of the sharp bottom. But there is 250mA variation here (1V peak to peak) you say. So whatever the bottom voltage is, that will tell us the minimum current reached. But we do know there is a ripple in the current of 250mA, which is a lot. It is possible you are hammering your LEDs with a lot more than that, given that the low point doesn't flatten out.
Can you read off the voltage at the bottom, there? That will give a little more information. (Assuming things don't change again!)
....
In the meantime, see about getting some toroid cores. Now here I'm out of my water depth. I don't know a good supplier for these and I'm not well versed on materials, generally. Here are a couple of sites that describes the various ferrite materials. (They are numbered.)
Personally, I'd prefer something called "high volume resistivity" and the highest mu you can get is is probably material #43. The high volume resistivity helps ensure you won't short things out with the core, itself. However, if you are using insulated wire and winding smoothly, I suppose any of the core materials are fine so long as they don't have high losses at the frequencies of interest. (On that score, if they rate losses at frequencies below 1MHz, you probably want to stay away from it.) This limits you to #43, #61, #64, and #67. The lowish permeability of #67 might be a bit restrictive, though. Here's another page that talks a little about the materials and something called AL.
The AL value is used for figuring out roughly what kind of inductance you are likely to get with some number of windings. The fuller formula for that is here:
A comment about the cores. Here's a page on their sizes:
An F-23 is just big enough to wind one layer of perhaps 30 windings using #32 wire and maybe 14 windings of #26 wire. Unless you get something with a large AL figure, you won't get too close to 200-400 uH with a single layer. You can always wind more, of course. Larger cores let you wind more windings on the first layer and the inner hole size is what will limit you on total windings, even if stacked.
Let's say you want to limit yourself to winding no more than 50 turns for primary and secondary, each, and that you want 400uH for each. Reading this again:
You can see that you want AL=1E9*L/N^2, if L is in henries. With N=50 and L=400e-6, you get AL=160 (or more.) So you need to find something around that value. Looking back at:
You can see that F-23 with #43 material fits the bill. The problem will be winding that many turns on something that small. That means fine wire and stacked windings, probably. But it would be tiny. An F-240 with #61 looks about right. But I bet they will be expensive.
Another possibility is to look at increasing the frequency a bit (reduces the inductance.) With L=200e-6, you only need AL=80 with 50 turns per winding. And so on.
Take a look at: (no recommendation here, I've never used them)
Anyway, finding the right core and the right supplier is a process.
Jon