[Mirror Adhesive][1] ("*mirror mastic*") exists:

![enter image description here][2]

As for how to hold it up; rest it on something:

- the top of the backsplash
- rest it on a strip of wood fastened to the wall

and use mirror clips while it sets.


  [1]: http://www.rona.ca/shop/~adhesive-mirror-adhesive-mulco-414642_mulco_paint_shop
  [2]: https://i.sstatic.net/sJwk0.jpg