Ok, so it's really just 2 states which need to be separated, right? Attached is a screenshot you can refer to, as I explain.
-- enable the Pen/Bezier tool
-- click once on the canvas where you want the line to start (notice that the line needs to start and end outside the state borders)
-- move the mouse over, and double-click where you want the line to end
I only guessed exactly where the new line needs to be drawn. I don't know how specific it needs to be, and I don't see any easy way besides 'it looks pretty close'. So if you have any further info about exactly where the new line should be, please explain. (I can think of "some" way to make it precise, it's just not exactly "easy".)
-- Object menu > Fill and Stroke > Stroke Style tab > Width (it looks to me like your new line needs to be 0.01 px wide)(to match with the rest of the map)
-- with the new line selected, Edit menu > Duplicate (or this button in the command bar
) (or there's a keyboard shortcut too)
When you duplicate, Inkscape puts the new path right on top of the original. So you probably won't notice that anything has changed.
-- switch to the Selection tool
and drag a selection box around the path that you can see, and it will select both -- look at the status bar to confirm (bottom edge of the window) it should say "2 objects selected of the type "Path...."
-- Path menu > Stroke to Path
-- deselect them (click on any open area of canvas)
-- click on the top new path, hold the Shift key down, while you click on California (Shift key allows for multiple selection) (in case you didn't already figure it out) (since the path is so narrow, it might be hard to select - try putting the mouse over the part of the new line which is over the ocean - you'll see the mouse pointer change when it's in the correct position to select it, then just click once)
-- Path menu > Difference
You might not notice anything yet, because the 2nd new path is partly covering it up. But repeat the last 2 steps, except this time select Nevada instead of California. After you Difference on Nevada, you should see the new white line between.
-- select either California or Nevada
-- Path menu > Break Apart
-- repeat for the other state
Now you can select the new sections of the states, and group them as needed.
Oops, I missed Illinois! But you can repeat the same process for Illinois.
Let us know how it goes