First, put the pot on a bed of stones in a wide tray with 1-2" sides. Fill that with water to just below the stones so it never gets into the pot. That humidity will help the brown ends. Keep the whole thing away from dry heat and from direct sun, but for as many hours of light daily as possible. Repot into a mix of half the present one (I can only assume it's coarse and gritty - normally desirable in plants that need fast drainage) and more soil, but not 'potting' soil, but something crumbly and light from the garden centre, that isn't full of peat, that compacts and makes it hard for the plant to drink properly (which may also be part of its present problem). And give it some time recover.

Never fertilize a sick plant. It just stresses them more.
You say the water runs thru so the soil isn't absorbing it so that is most likely the cause of your problems. It's probably dying of thirst with the hard compacted soil. The water never gets to the roots.
I would repot it with new well draining soil that does not have fertilizer in it. Make sure the new soil has perlite in it or some other substance to promote drainage.
Peace lilys like to be potbound so no more than an inch between rootball and inside of the pot.
As Lucy said your soil is probably mostly peat and hard and compact. Once peat gets hard it's very difficult to soften. You can soak it in a pail to soften and remove it or break it apart. Just be careful not to damage the roots.
The plant is also really stressed from all the changes you have made.

Actually it's not true about fertilizer and sick plants, though most people believed it (along with a lot of other now-proved-wrong ideas) for a long time. Certainly stress can be added if a plant has been neglected (or over-babied) and not fertilized. As long as you apply reason and a light hand, it won't hurt anything and may even help.

Sounds like a watering problem.If the leaves are drooping and the water runs right through, then the water is probabay not soaking the entire root-ball. It could be very root-bound but you'll need to ck, if so, it needs the next pot size up.

If you bring your peace lily over the kitchen sink and REALLY pour the water on (tepid) getting the soil soaked and the leaves perk up, then probably a bigger pot is in order (one size up)
If the leaves DON'T recover, it's probably been over-watered at some point, maybe needs a smaller pot with better drainage, rotted root sections cut away if it's not too far gone.

Billy Rae

" -the soil isn't holding water well...it just flows right through."

This is surely your plant's problem. Soak the entire pot in a pail of like warm water until the soil is completely soaked, allow to drain before returning to it's place. Never let a peace lily completely dry out. These plants love water. Mine spends the summer in a small 150g pond in the shade.

Something I have had in "My Favorites" and whole heartily agree with.
Some may not.

Nutrient deficiencies are often assumed to be the cause of a plants ailments and fertilizer is assumed to be the cure. Nutrient deficiencies are, in fact, the least common cause of all plant problems. Indoor plants use very little nutrients and they cannot be force-fed. Most potting soils have several years worth of nutrients. If your plant has been in the same pot and soil for several years or more and is growing well, then it may benefit from fertilizer.
"Ailing" plants are poor candidates for an increased nutrient supply.
What needs to be addressed is the real reason for the plants illness and then be corrected.