I have created this image, I couldn't manage to make the corners look rounded (see below)
How can I make the corners round in Illustrator?
I think you should redraw it, because it would take longer to fix than it would to redraw.
When you draw your line, don't worry about the curves, just make straight lines with normal corners and apply a thick black stroke.
Now use the Selection Tool V to select the entire line.
Hold down CTRL and you will see the Live Corner circles appear.
Click and drag on these to round the corners
On the corners where you don't want rounding, click the Live Corner circle for that corner, then click and drag it out to make it a regular corner.
You can also use the same trick on the big black rectangle.
As Vincent mentioned, Billy's answer is great, but only for the newer (CC) versions of Illustrator.
In the older versions (namely CS6) you can just go to Effect → Stylize → Round Corners
You can also create the shape (as per billys suggestion) and then apply a thick stroke, then set the stroke to round the corners.
You can now go to Object → Expand to turn the stroke into a regular path
Before you round the shapes, using the Direct Selection Tool A select the exact anchor point(s) from the corner(s) that you don't want rounded and copy them (Ctrl + C)
Now make the corners rounded, and paste in place (Ctrl + F)
(Once you do Expand you can use the shapebuilder or pathfinder tools to merge them together)