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There are probably a few different ways to approach this. Perhaps the hardest would be to use File menu > Import, and import the JPG. Then open Path menu > Trace Bitmap. That tool is a lot to handle for a new user, but I'm thinking to use the Colors option with 4 scans. That
should result in a very close approximation of your JPG. I'm not on my computer with a current version of INkscape right now, but later I'll be able to experiment, if that doesn't work well.
Another way is to try the "manual" trace with the Pen/Bezier tool again. First, import the JPG (File menu > Import). Next open the Layers dialog (Layer menu > Layers...). Add a new layer above the original layer. Select the first layer with the JPG in it, and reduce the opacity to some comfortable level. If you don't want or need to do that, you can skip adding a new layer.
Next, zoom way in until you begin to see the pixels in the JPG. Now engage the Pen/Bezier tool

. On the tool control bar (across top of canvas) click "Create sequence of straight line segments" button. Find a starting point and click once to start a new line. Move some distance along the border between color and white, and click again to set the next node. Be sure to set all your nodes either over the darkest pixels, or the lightest. If you use the darkest ones, your shape will be very slightly smaller than the original and if you choose the lightest, it will be very slightly larger. We're only talking about a few pixels difference though. You could choose to place the nodes over the middle between lightest and darkest, but I find that difficult personally. But whatever works best for you.
Now, when you clicked once to start the new line, did you notice the tiny square? When you come back around to that starting place, be sure to double-click on that square. It will turn red when the mouse is positioned properly. This might be why you couldn't add color to your first attempt -- the path might not have been closed. OH, and note that when you first open Inkscape, the page is at 35% zoom level, where it's hard to see whether the path is open or closed. So that might have been part of the problem with the first attempt as well.
After you finish drawing and close the path, you'll need to convert most of the nodes. Because you're drawing in straight line segments, you probably see some corners where you don't want to. So engage the Node tool

and select one of the paths. Next do Ctrl + A to select all the nodes. While holding Shift, click the node at the sharp point of the....I don't know, feather-like objects. This deselects that node, while leaving the rest still selected. Now look again at the tool control bar, and click "Make selected nodes smooth" button.
Now you should be able to fill the paths with color. And you can remove the stroke if you don't need it.
As you might have noticed, there is another way to use the Pen/Bezier tool. If you're not in 'straight line segment' mode, when you click and drag, it makes a smooth node and a curve. Since I often find it hard to click and not drag, especially if I'm in a hurry (and I also find it hard to control the curve in this way) I've found it much easier to use straight line mode, and then later convert the nodes to smooth. I suppose others are more skilled than I, but that's what works for me

The 3rd way I'm thinking of is perhaps the easiest. Import the JPG. Engage the Paint Bucket tool

. Place the cursor over one of the shapes and click. It will automatically draw that shape (by estimating the difference between colors). Do that for each shape. Then you can use either the palette along the bottom of the window, or the Fill and Stroke dialog (object menu) to change the colors. The problem with this method is that Inkscape can't draw the shapes as precisely as you. If you zoom into a corner, you might see several nodes actually forming a sharp curve, rather than a sharp corner with one node.
Let us know how it goes
