What is the border state that Lincoln kept with the Union thereby saving Washington D.C?

He allowed Kentucky to stay neutral for the first year of the war.

Maryland was a bigger worry - if they had voted Confederate, Washington would have been totally surrounded by enemy states. So Lincoln jailed some pro-Confederate politicians, actually breaking the law of habeas corpus.

Missouri was the scene of bitter guerrilla fighting throughout the war, and Lincoln never really controlled it, but at least it did not become a Confederate state.

Delaware was too small to make much difference. It raised regiments for both armies, mainly Union.