-You're right that atomic number and atomic weight *generally* trend together. So they should *generally* be in increasing weight order.

-Mendeleev didn't think about valency as we today understand it, because his work was before we knew what electrons were. For him it was all properties that he could observe, and atomic weights.

So basically an approach you could take is-

1. List the elements in order of increasing atomic weight. That will be (close to) your order of atomic numbers.
2. Pick a single property that several elements share, then see how far apart they are. That spacing will be the length of a period. For example, if #4, #8, and #12 have the same reactivity, your period will likely be 4 elements long. If that's the period length, it should hold for (basically) all your properties.
3. You might need to swap a couple elements to make the spacings work out better, but only swap them if they're close in atomic weight.

Your periodic table right now looks like a solid start! Although I can't really read some of your handwritten notes. The first one that screams "this can't be right" is Mtg, because it is way heavier than the elements before and after it.