YIKES! Those roots! I would say it is desperately needing a bigger pot and maybe a root trim. This is probably why you're having watering problems. I would get a 10 inch pot, tease the roots a little bit then pot up in a well draining soil. Now is really not the time to be playing with roots, but you do want to loosen the circling roots at least so they will grow into the new soil. OR you can wait until it's a more opportune time to repot and keep an eye on watering until then. Water when the leaves just start to droop but not that bad.

Bigger pot..., in the meantime..., more water!

I know it doesn't seem possible, but a 10" pot holds about 5X (4.8) as much soil a 6" pot, so be careful you're not over-potting.

How I would treat the plant:

First, I'd make sure I have a slightly larger pot ready to go with screens over the drain holes and a 1" layer of potting soil mixed 50/50 with perlite covering the bottom of the pot. I'm only recommending that soil mix for this application because it should be compatible with the soil the plant is in. I'd prefer something with more aeration that drains better than that, but you shouldn't use 2 soils that are significantly dissimilar in one container. I'd then use a utility knife with a fresh blade or a new single edged razor blade to cut 4 vertical slits 1" deep in the root mass from top to bottom, then I'd cut or tear off (with my hands) the bottom inch of the root mass, set the plant on the bed of soil, and fill in around the perimeter with fresh soil and water lightly. After a couple of waterings, I'd fill in again around the edges, making sure I don't add enough soil to cover the top of the original soil/root mass. Don't worry - your plant won't balk at this treatment.

Alternately: you could just hold off until the plant starts telling you it wants to grow, (prolly around Easter) then divide it with a butcher knife or similar tool, pot the divisions, and share your largess.

Al

Go with what Al said but I must admit, Al's more brave than I am. I hate cutting roots and only do it when It is absolutely necessary. IF you can tease the roots without cutting them you could easily fill a 10 in pot with those roots. otherwise, if you're brave, trim the roots and go one size smaller.

I recently posted about repotting my peace lily. I washed all the old soil off, trimmed about half the roots, and repotted in a larger pot (I always like to go larger). I plucked off more than 100 leaves (forget how many). I used a gritty mix: 2 parts turface, 1 part pine bark. My plant is beautiful. Never suffered

http://forums2.gardenweb.com/discussions/3606558/peace-lily-in-gritty-mix-root-pruning?n=9

Thank you everyone for the super useful advice!