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Sounds like a fabulous idea! Yes, you can glue one mirror atop another. A common oil-based construction adhesive such as "Liquid Nails" will work fine. I would suggest using a caulking tube rather than a can of glue... it's much easier to apply and neater to use. Have some paint thinner handy for cleanup, though.
The big problem is keeping the new mirror in place on the wall until the glue dries! While the glue is wet, some sort of support will need to be improvised to keep the mirror from sliding out of position.
I'll make a suggestion as to how I might do this, but don't feel constrained by it. Most home repair problems have many solutions. So look at the situation "first hand" and use your imagination to improve on it! Make a strip of plywood or other wood that will sit on the vanity top and support the bottom edge of the new mirror. One continuous piece would be best, but two pieces (one at either end) would also work. You might have to use two pieces (or notch out a single-piece) if the sink is in the way!
Test-fit the mirror and the piece of wood. If you are satisfied with the position of the mirror, tape the wood to the wall to hold it in place. A light-tack tape such as "painter's tape" should work fine, but you will have to be careful that the mirror does not sneak behind the wood or it will slide down!
Now to the gluing technique. Since neither mirror is porous, you can't just put the adhesive on, press the mirrors together and expect a great initial "tack" or holding power. Instead, you must use the same technique that is used when installing wall paneling. First, apply the glue to the mirror on the wall at least within an inch of the imaginary border of the new mirror. Apply a 1/4" bead at least 2 inches from the perimeter (you don't want any squeezing out) and a few vertical beads 4 to 6 inches apart. Then press the new mirror in place exactly where you want it to be, resting on your wood base. Immediately pull the new mirror an inch or two off the wall from the top and allow the adhesive to air dry for about a minute. Then press the mirror back in place. Now the new mirror should really grip to the old one.
I wouldn't trust the glue 100% for at least 24 hours. Use whatever you have available to keep it against the wall. Since it will be on the mirrors, not a finished wall surface, a stronger tape such as duct tape can be used. Don't seal the entire edge of the mirror (you'll slow down the drying of the glue)... just a few strips along the top and sides. You could also lean something heavy against the mirror. Use your imagination! Again, the mirror will tend to stay in place but you want to be absolutely sure it doesn't move! Once the glue dries, the mirror will be in place forever.
Now, there is one potential problem. Though unlikely it might happen that the supporting wood will not dislodge easily if it is wedged between the mirror and the vanity top! If you think this might happen, just insert a ©" wood shim or strip underneath the support (take the thickness into account when measuring or your mirror will be too high). You can even tape it to the support if you want. Then, if the support is difficult to remove, it can be separated from the shim and should then pull out easily!
Thanks for the tip. Makes sense to me! I'll see it gets added to our Adhesives and Glue page… https://www.naturalhandyman.com/iip/infadh/infadh.html.
Another great suggestion! Bless you for sharing it, because I feel a little ignorant on these glue storage issues... mostly because I carry everything with me in my truck so refrigeration is not part of my daily routine! In fact, keeping glues from not freezing is my biggest concern in the winter months.
Goop requires air to dry and requires at least 24 hours to gain full strength. Goop does not "set" like an epoxy... it dries by evaporation of a solvent.
Hey... don't beat up on yourself! If your repair lasted a year, you probably did nothing wrong. It is important to note, though, that using a vinyl patch along with an adhesive helps to reinforce tears and holes, thus extending the repair's life. I am not a vinyl repair expert, and it may be possible that Goop is not be the best repair product. Only your owner's manual would know for sure!
Vinyl patch kits are commonly available in hardware, home and pool supply stores.
In speculation, if you were to use a vinyl patch with Goop, I would recommend applying the Goop to both surfaces, press the patch in place and then pull it off for a minute or two to allow the Goop to begin to set. Then press parts back together. This tip is from the Goop instructional material for bonding non-porous surfaces. As mentioned earlier, the solvent in Goop needs to evaporate for the Goop to set... allowing this "pre-drying" speeds the process along!