Use the existing rails and replace the missing screws. There should be holes from the screws in the mirror frame and the dresser. Just make sure the screws are at least 1/14" long.
I agree with William (as usual!)
Head to your nearest hardware store and get more screws. Be sure they at least 1-1/4 to 1-3/4 inches Long to replace missing screws.
Done that. Hard, so far, to find right diameter that fit the bracket hole and weight. (I haunt the hardware stores)
There are so many different size screws available, something has to fit. Check out Home Depot, Lowes, or Menards. I don't understand what you mean by weight.
Was the mirror originally from this dresser? If not you may have to try a different approach. Otherwise a #10__1 1/4" to 1 1/2" Phillips flat head screw usually does the trick.
It is the original. I'll check that size, thanks