Delete part of a stroke of a shape

copy your shape, then paste in front and set the fill to none, and stroke to black. Then, use your scissors to cut the black stroke areas that you want to delete.

The bottom shape that you copied from can be a fill with no stroke.

KB,

Your images seem to suggest certain lacks in the coherence of your shape, which seems to consist of (at least) two objects, an upper left one being a separate path extending from the very leftmost corner where there is an obvious meeting of two open end (Butt Cap) Anchor Points and to somewhere between the two cutting points in the third image; in the third image you can see the two stretches of unstroked boundary line between the very leftmost corner and the rightmost cutting point: such an unstroked boundary line proves the openness of the path(s) to the lower right; and the leftmost part of it also hides part of the upper left open path which dips down behind it, showing that the upper left open path is beneath the other(s) in the stacking order. You ought to be able to see the different paths in an expanded view of the Layers palette.

Closing the shaping into one path would make the whole thing more sound, it ought to be possible to just select both/all paths and Ctrl/Cmd+J.

As an alternative way of cutting the stroke (with just one path forming the shape), applicable (almost) regardless of version, would be to select the path and Ctrl/Cmd+C+F and then remove the stroke on the original and remove the fill on the copy, then cut the latter as desired and delete the unwanted part.

Draw two additional lines where you want to "cut it". Then select all the objects and use "live paint".