Quick and dirty? All you "have to" do is stand the harp, preferably having removed the mirror first (just to make it easier to handle), EXACTLY where you want it to be on the dresser top, then mark the edges front, back, and both ends with painter's tape on the dresser top. (Painter's tape won't leave any residue on the surface when you remove it.) Then lay the harp down in front of the tape and mark the tape so you know where the screw holes (four of them, two with screws, two empty) are on the bottom of the harp.

You'll want to use three wood screws to hold the harp to the base, one in or near the middle and the other two out toward each end, but spaced so as to NOT come within an inch of either of the screw holes on the bottom of the harp and at least 1-1/2 inches from either end so as not to split the end.

You'll need to measure the thickness of the dresser top and the thickness of the base of the harp (the part with the holes) because you cannot use a screw longer than two thirds to three quarters of that combined length.

Using a drill bit SLIGHTLY SMALLER than the diameter of the wood screw you'll use, drill holes for the three screws from the top down, then, holding the harp in place either with clamps or with the assistance of a strong and steady-handed helper, drill back up through those same holes in the top, into the base of the harp. This is the trickiest part because the harp has to be held absolutely still and the holes cannot be too deep. You measured the length of those combined thicknesses, so reduce that length by 1/4 or 1/3 and mark your drill bit by wrapping a piece of tape around the bit. When the tape hits the wood, stop.

Remove at least two drawers for work space because you'll being screwing from beneath the top, up into the base of the harp. If you have them, you can use power tools to drive the screws most of the way but do the final tightening by hand. It's too easy to overtighten and risk cracking old, dry wood when using a power tool.

This is NOT a restoration procedure, it's merely a way to make your dresser more usable by holding the mirror on. I presume the value of this piece is primarily sentimental. If I'm wrong, if this is an antique of significant monetary value, don't even consider "fixing" it in this way; it would seriously diminish its value to a collector.

If you wish to "restore" the piece, let me know. I'll tell you what needs to be done, but it will be neither quick nor easy.

PS Just for starters, the little disc you've pictured twice is a Panel Nut. It's thin, it's threaded, and PRAY THAT YOU HAVE BOTH OF THEM because if you have only one it's going to be nearly impossible to find another one just like it, although there are ways around that.