As nice as it is to have a community of like minded fellows all adding input, unfortunately the advice seen here is from people without any real personal experience.
My shop provides an interesting niche service to repair professionals around the USA.
We do in fact rebuild these so called "impossible to separate" assemblies.
I am giving this advice freely as a professional who performs this task numerous times every single day, not as a novice tech who dabbles in phone repairs.
There are two ways to properly perform this task on any OEM adhered lcd assembly. One way is very costly to setup and the other is not but it is slightly more time consuming.
I will not go into the expensive setup we have here that would cost you thousands of dollars, because on your level it is unrealistic. The good news is that you can separate the layers YOURSELF using nothing more complex than a simple hair dryer and some odd and end trinkets.
Following these steps you will have a reconditioned lcd assembly in under 30 minutes using common household items and a few you will need to hit the harware store to obtain.
Hot Plate Or Griddle
One Deck Of PLASTIC COATED Playing Cards
One Roll Molybendum Alloy Wire
Safe Scrape Plastic Razor Blades
50 ML Tube Of UV LOCA (Ultra-Violate Curable Liquid Optically Clear Adhesive)
LOCA lcd mold designed for the lcd being repaired.
Isopropyl Alcohol (90% or better)
UV Nail Dryer
Do not listen to what was said earlier about using no heat. In order to properly loosen the adhesiv you NEED 100 degrees F (77 C) or you run the risk of destroying the polarizer layer.
Preheat your griddle to 100 degrees Fahrenheit or 77 Celsius.
Once preheated place the screen assembly FACE DOWN on the griddle.
Set the thermometer somewhere handy and point it at the BACK of the lcd layer.
while the lcd is coming to temp (100/77), prepare about 2 feet of Molybdenum alloy wire.
your screen should come to temperature in about one minute depending on ambient temperatures in the room at the time.
Its Time To Make That Magic Happen!
While wearing leather gloves, wrap the ends of the wire around your index fingers on each hand enough times to give you a very sturdy grip. Keeping the assembly face down on the griddle apply slight pressure on the lcd with your thumbs while you gently work the wire between the glass and the digitizer/polarizer layers. You only want to get the wire in a few millimeters, just enough to get your playing card in behind it.
Still with me? GREAT!
Take off the gloves, leave the wire in the glass and slide a playing card in right behind the wire. Make sure the card spans the ENTIRE width of the lcd and protrudes from both edges, left and right.
Turn off the griddle and using one glove gently but steadily pull the wire out from one side of the assembly.
You are now 5 minutes into the removal and are almost in the home stretch! Here comes the "hard" part.
Have someone hold the hair dryer for you and point it at the glass, to the very bottom edge of the card. Moving together, slowly but steadily work the card down between the layers, separating the glue. Do not bend the card towards the lcd at this point or you may shatter it. Continue moving the card through as your friend follows the edge of the card down with the hair dryer. You will more than likely need to throw away the card and use another 3-4 times to get all the way through as the card will get mangled and lose its edge, becoming ineffective.
In ten or so minutes you should be all the way through! if so congratulations you are in the home stretch and the hardest parts are over.
You will notice that your lcd is free of all the glass but is also covered in gooey gum. Much of this can be removed by simply grabbing some tightly and peeling it back with your fingers. If you run into any stubborn gum use the safe scrap plastic razors to remove it. Do not use any solvents to remove the gum unless you are proficient with a soldering iron and do not mind re applying the back light layer on the rear of the lcd.
Got it all off? GREAT! This next section is CRUCIAL to pay attention to unless you want air bubbles in the layers so please read many times before you start the process, because once you lay the glass down you can NEVER lift it back up again without trapping air.
Place your lcd face up in the LOCA mold. Apply a line of LOCA straight down the middle of the lcd from top to bottom. Excess will not hurt but it is messy, however you are better off using too much adhesive than not enough. Once the LOCA is applied, place ONLY the bottom edge of the glass into the mold but hold the top edge up and do not allow the glass to contact the glue yet. VERY SLOWLY begin to lower the top edge down to the lcd. ONLY move as fast as it takes for the glue to make it all the way to the left and right edges, it will like watching a horizon of glue rise to the top and outward to the sides.
Once it is layed down and you are happy with the placement, put the entire LOCA mold, lcd and glass, into the UV nail dryer for 3-5 minutes.
Congratulation, you have rebuilt your lcd assembly. In our shop we often do have to perform this task in a similar fashion as the machines we use do not work when the glass is completely shattered, so this advice is based on my personal experience from owning one of only 3 facilities in the United States that is capable of this on a large scale.
I recommend trying on a cheap screen like the iphone 4 a few times prior to attempting say, an iPhone 5c or Galaxy S4 for obvious reasons.
Keep in mind that iPhone screens are so inexpensive comparatively speaking that there are probably many things you could be doing with your time other than attempting this. But if you are like me, and it always bothers you when someone tells you something cant be done, then enjoy. Remember, patience is the key. Frustration, shaky hands, being irritated or distracted often leads to a shattered lcd. If you are having trouble put it down and take a break for 5 minutes.
Best of luck everybody!
quick edit: I forgot about the Alcohol! using a q-tip NOT dripping in isopropyl alcohol gently clean excess glue form underneath your new glass. Unlike solvents, this only removes uncured LOCA and will not effect the bond you created. Be careful to use as little as possible or you may need to solder a new back light layer on the lcd, as even alcohol is not pure and contains some water which will stain the otherwise immaculate back light layer, giving you that bad whiteout splotchy look.