Melatonin: The regular form helps me get to sleep but not stay asleep; the long acting form helps me get to sleep and stay asleep longer. I do better with lower doses. I take 1mg of Natrol Long Acting melatonin at bedtime.
L-Tyrosine: this is great for helping me to sleep better and is one of my essential supplements. I take one 750mg capsule at bedtime. 750mg works better for me than 500mg.
L-Theanine: this is used in a lot of sleep supplements. At one point I could tolerate it, but now causes problems such as feeling drugged for a couple of hours after waking. Some people have nasty reactions to l-theanine (stomach pain, vomiting) so it's a good idea to start with a low dose.
I've read comments from people with CFS/ME who get sleep studies and most people found them to be useless unless sleep apnea was found. The sleep disruption experienced by people with CFS/ME is something that most doctors don't understand so they just throw pharmaceuticals at the problem which can introduce nasty side effects such as tolerance, addiction or benzo withdrawal. I've been able to avoid pharmaceuticals so far by relying on a cocktail of supplements.
I would try the l-tyrosine first. I know within a day or two whether a sleep supplement helps but a trial of at least a week is useful for many people, especially if useful effects aren't noticed right away.
My current nightly sleep cocktail is: 1mg Natrol long acting melatonin, 750mg l-tyrosine, 1 bag of Nighty Night tea with valerian, 350mg of magnesium citrate, two tsp of apple cider vinegar, and a couple of spoonfuls of oatmeal (helps to maintain blood sugar during the night and produces serotonin). Other daily supplements that help my sleep include iodine, methylfolate, and LDN.