They are indeed a mess. Look for the growing tips which will have some new or small growth on them. Go down from the tips and cut a chunk of the roots off around the growth tips. Discard the rest of the roots. What you want are the growing tips with some roots extending from them. Repot with soil and gravel over the roots, leaving the growing tip up and dirt free. On an established plant, you will have multiple tips so you can divide the lilies too. Typically, they are really hardy. As long as you protect the tips and have some roots extending from them, they are easy to repot and virtually indestructible. Once repotted and fertilized, plunck them back underwater and let them go. They should be fine! Good luck!

I've learned that hardy waterlilies are pretty darned hardy. I usually just start cutting things apart with a saw and keep the piece that looks the best.
This spring, mine looked like it had a couple pineapples growing out of it! I didn't even dump the plant out and sift through it. I just cut big portions of the plant away and filled it in with more dirt.
I do keep the stuff I cut out in a bucket of water for awhile, in case I chose the wrong portion to keep. But its always worked out. Then I throw the saved stuff on the compost pile, when the lily gets going.
I don't think you have to be as careful as you think you do.
Good luck!

Oh, thank heavens and thanks for the replies. When I tipped the pot, and seen what I had, I thought "Oh, gotta get on gardenweb and ask the experts." : )