Brian is right, Levemir doesn't last 24 hours for the majority of people. If that is what is happening to you, then it makes sense that it would be fading out during the time you are sleeping. You should discuss this with your doctor.
Another thing that produces superior results—again, for some people—is to split the Levemir dose in two (some at night and some in the morning). Since two cases of diabetes seldom behave in precisely the same way, trial and error is a big part of determining the optimal strategy for you. But based on what your CGM is telling you, Brian's suggestion is probably the one to try first.