Not to state the obvious, but the League of Nations was the precursor of the United Nations. So 'naturally' they amount to the same thing in Bible prophecy.
Yes but in Daniel 11:31 the "disgusting thing that causes desolation" supposedly refers to the "setting up" of the United Nations in 1945 (not the setting up of the League of Nations in 1919). Daniel 12:11 is clearly referring back to that event in Daniel 11:31. Therefore when Daniel 12:11 says to count 1,290 days from the "setting up" of the "disgusting thing that causes desolation" it must be referring setting up of the United Nations in 1945, not the League of Nations in 1919. And that would still be true even if we say the League of Nations was also the "disgusting thing that causes desolation", in which case the "disgusting thing that causes desolation" was set up twice, in 1919 and 1945, but the most natural interpretation would be that Daniel 11:31 and 12:11 must both refer to the same event.