I agree with the soil being a BIG part of the problem or THE problem. You definitely need some different growing mix...optique's recommendation of a 50:50 peat moss and perlite mix would work great.
Personally, I always rinse my mix ingredients...seems to help with getting rid of "milk" from the perlite and reduces the amount of mold and mildew that can grow on the peat moss. But, many people don't worry with rinsing it. DO NOT BUY any Miracle-Gro ingredients!!!...they are full of fertilizer which is deadly for many carnivorous plants, especially flytraps and sundews.
Your pot "looks" kind of like similar terracotta/clay pot, at least by color. Plastic is the best with the least issues. There are also resin pots that are thicker than regular plastic ones, but much lighter than the clay pots. I feel that the plastic pots are "tougher" than the resin ones. If your pot felt heavy before filling with the potting soil and appears to be 1/4" thick or thicker then it's probably a safe bet to say that it's a clay pot...which is bad because it will leach minerals into the soil. Sturdy plastic is the way to go, IMHO.
Looking at all the rough stuff in your pot I agree it's some kind of coarse potting soil. If it was me, I would:
1. Get some good grow mix. Moisten it. If you rinse it that will cover the moistening part. I rinse the two ingredients separately and then combine them. Set aside.
2. Remove the plant from the pot and rinse its roots in some distilled, RO, or rainwater that you've put in a large cup or bowl. Kind of slosh it around in the water, letting the old dirt fall away. You might want to empty the dirty water and refill with clean water a time or two.
3. Now, if you have some LFSM use it to cover the drain holes to keep the grow mix from escaping.
4. Put about an inch of grow mix in the bottom (over the LFSM if it's there) and firm it down well.
5. Fill the rest of the pot with the mix...all the way to the top edge of the pot (it will settle). Firm this mix down a bit but not as firmly as that one inch in the bottom.
6. Make a deep hole in the middle of the grow mix with your finger, a butter knife, whatever.
7. Carefully drop the roots of the flytrap down into the hole, trying to coax them straight down if you can.
8. Naturally you'll need to firm the grow mix back in around the flytrap. You want a small amount of the "white'ish" part of the leaves (the part at soil level) to be sticking out of the grow mix. If you can't see a bit of white you've planted it to deep.
9. Top water the pot to settle the grow mix in around the flytrap and its roots. Let the water drain through and throw the drain water away.
10. Check to be sure the "white" is showing above soil level.
11. Add a 1/4-1/2 inch of water to the water tray.
12. Be patient.
In regards to the physical part of placing the plant's roots into the grow mix you can do it a different way, too. You can't replace 5, 6, and 7 with this: Lay the pot on it's side and fill it with grow mix half way. Then lay the flytrap on your hand with the leaves pointing towards your arm and the roots stretched towards your fingers. Insert your hand into the sideways, half-filled pot and roll your hand, letting the flytrap fall onto the grow mix. Now using a large spoon or something fill the upper, empty half of the horizontal pot with grow mix. When you have it well-filled with grow mix turn it upright and finish firming up around the flytrap. Again, making sure you see some of the white'ish, lower parts of the leaves above the soil level. Proceed on with #8, etc.,.
This is just my take on the situation. The plant may be too far gone to revive or it may come back nice and strong...but, it will take a long time if it does rehabilitate. If you can, you might want to get another flytrap and start it off in good conditions. A good looking plant will help you stick with rehabbing the sick one.
I never did see what type of lighting you have it under. Has it had any kind of lighting other than window sill lighting?
I tend to ramble a bit at times...I hope some of this might help.
Best wishes...