It's not a case of giving it "back" because it didn't belong to Canada. The Northwest Angle is part of the US because of inaccurate maps and assumptions made during the negotiations with the British to end the American Revolution. The peace treaty, which was based on the erroneous maps, says the border between the US and the British possessions to the north would run "...through the Lake of the Woods to the northwestern-most point thereof, and from thence on a due west course to the river Mississippi...." Problem is, the Mississippi doesn't go that far north. Its origin at Lake Itasca was not known at the time, and it's south of the Lake of the Woods. Meanwhile, the northwestern-most point of the Lake of the Woods is in a narrow arm to the north of what we now know as the Northwest Angle. This was all discovered by 1815, so in 1818 the border was renegotiated. The provision about the northwestern-most point in the lake was kept, but from there the border ran (and still runs) due south to the 49th parallel, at which point it turns west. (A later treaty relating to the division of the Oregon Country provided that it would deflect south near the Pacific so as not to split Vancouver Island.) The use of the 49th parallel led to other US enclaves accessible only through Canada —Elm Point, Minnesota, and Point Roberts, Washington, are the main two.
