I can see how struggling to repeat your mantra perfectly until you fall asleep might keep you awake; perhaps consistent recital of it isn't as important as getting to sleep might be. Along the same lines with what MisakaMikoto already advised above, you might consider making your mantra more a backdrop to falling asleep, rather than your primary focus.

This might sound counter-intuitive, but it won't be if you remember that going to sleep is the primary goal for lucid dreaming, because if you are not sleeping, you are not dreaming. So you might just relax, even go about going to sleep as you normally would (as long as "normally" does not include distractions like phones, TV, or computers), and let your mantra sort of repeat itself softly behind you as you let yourself fall comfortably asleep.

Your mantra, if it is a meaningful one, will tend to take care of itself as you move closer to sleep. And, especially with MILD, if you have set your intentions well enough, it really is okay to slip up and skip a few repetitions.

I hope that helped, or even made sense...