I would...
Use a glass cutter to run scores across the face of the mirror in a roughly plaid pattern. The idea is to give the mirror some encouragement to crack when you start to pull it off. Smaller pieces are good to handle. Do not run the glass cutter along the same score multiple times. One firm score is enough.
Make the scores no farther than 6 to 12 inches apart. Now, tape up the entire mirror. Packing tape would work, as would masking tape. Use enough tape to stop any pieces from falling down and shattering or hurting someone.
Now, use a hammer or pry bar to lever up under one corner of the mirror. As you pry out slowly, tap with a metal object (a small hammer is good, but anything will work, even a heavy screwdriver or the back end of the glasscutter) along the scores you made with the glass cutter. The glass will probably start to crack now along the score line you made. Break away the mirror in small pieces so that you can carry it away easily.
Be very careful of the edges, as they will be sharp. Wear good gloves as well as eye protection. When you are done, sweep up the area carefully, then vacuum. Make sure there are no slivers of glass on the floor.
I'm sure there are several ways to secure your mirror, However the two basic methods are 1) mechanical side clips and 2) Mirror adhesive.
The safe and easiest way is to use mirror clips that screw into the wall around the parameter of the mirror. Small drywall anchors are normally used if you're not lucky enough to have a stud along the horizontal sides. Try to find a stud along the top or again, use anchors. Be sure to read the ratings on the clips to determine how many you need for the weight of your mirror. It is really your call if you want a gap under your mirror. personally, I would leave and inch or two for cleaning purposes. A mirror resting on the back splash may collect water and dirt.
The second method is to use a special mirror adhesive. This makes for a very clean install, but can be very tricky. Read the directions on the adhesive completely. Be sure you have a firm finish on your wall to apply the glue to. Some side or top supports may be needed to hold mirror in place until the adhesive has cured. Once you use this method, it is semi permanent as it will normally destroy the sheetrock when removed. CAUTION!!! Only use a mirror adhesive if the mirror manufacturer recommends that method as an option. Some mirrors will not glue up and the reflective silver layer will pull off and allow the mirror to come crashing down!
Best Answer
3 possibilities that I can see.
Tape every square inch of it and just start smashing. Wear serious eye protection and a face shield, gauntlet gloves, heavy clothing, head covering, ... in short, don't have any exposed skin.
Tape, protect yourself (as above), and try to get a pry bar under the edges. You'll need to do a bit of drywall repair, but you might get lucky and not have a pile of shards.
Buy (rent?) a specialty suction cup slide puller. (This is scary at the best of times and probably unavailable at normal rental...) Edit to add youtube link: https://www.youtube.com/watch?v=dQne8Caw1LI