When you water it, do you flush out the entire pot? Or do you only give it small amounts, so that it doesn't drain through? It looks like it's not getting watered properly, causing mineral buildup within the soil. That's most likely why its leaves are turning brown along the edges. Judging from your photos, the soil looks way too dry.

If it were my plant, I would soak the entire pot in water all the way up to the soil line. After about an hour, I would drain the water away and then flush it with plenty of fresh water until the murky water runs clear out of the drainage holes. This would help to remove most of the minerals that have accumulated within the soil.

This, however, is just a quick short term fix. I would repot it with some fresh soil as soon as possible; using a fast-draining soil that doesn't stay soggy, such as a mix designed for cacti & succulents.

With the sun getting lower on the horizon, I would move it closer to a window that receives plenty of filtered sunlight. It needs to gather up as much energy as it can to recover, and it can only do that thru photosynthesis.

A pot that's roughly the same size would do just fine. They don't have to be rootbound to be happy, and it's not currently over potted either. Your plant is still relatively healthy. There have been some that were far worse that have bounced back from near death.

I think leaves will drop on their own if the plant decides it can't support its foliage with the roots it has left. If you decide to cut any leaves, make sure to sterilize your cutting tools with rubbing alcohol-rub the blades throughly and let dry. Personally I would leave the plant alone and see what it decides to do. If leaves do drop, the plant should be able to reabsorb some nutrients from the leaves before they fall.

You have a beautiful peace lily. I hope it recovers soon!

Cityplants, be sure to read the thread at the link Al gave you.

Once you have a handle on root health and how overwatering affects them and take some steps toward meeting them happy, you'll be amazed at how much your plant can thrive even before you repot it in good soil when the time is right. I certainly am, even though I won't get to repot until June!