It is entirely possible that if the toilet was running hard enough that that is the source of the problem. Assuming the weather conditions are fairly dry and surrounding ground is not saturated, I'd give it a day or two to subside and let the leach field catch up. If the problem doesn't clear quickly with the toilet fixed, I'd call for a pump and have them check the backflow of the leach field to determine if there is blockage downstream from the tank. Good luck.
I remember Neillsville from "The Rock" radio station. ;)
So, to answer the question, we need to know where the water is coming from. You mention that it's due to a high water table.
As such, I'd not ever finish the basement. It will always be one power outage or one clogged drain away from being ruined again.
It also means that your dehumidifier will never 'win'. Basements are moist by default simply due to cooler temps, but if you have constant hydrostatic pressure, it will always be an uphill battle.
Your (what I call) interior footer drain will certainly help, but you're at the mercy of the drains always remaining unclogged and the sump pump always having power. You'll also have to make sure it can handle excessive rain during those big midwestern downpours.
If it was merely an occasional rain/snow melt issue, I'd feel OK finishing the basement, but given the high water table, I'd consider rethinking doing a full finish project down there.
If your water problems are primarily at the footer, It may be OK. As that is exactly what the drain system is designed to handle. But if you have actual cracks in the walls and water is coming in there, I'd be definitely be hesitant.
One other note...a bit further west into MN, the clay soil we were on had a lot of radon. So you may want to get a radon test before thinking of finishing the space as well.
Oh, one last comment: note that the systems being proposed are in no way 'waterproofing' systems. They are water abatement systems--in that they are designed specifically to not be waterproof and actually allow the water to come in so it can be moved out. It's a nit-pick, for sure, but actual waterproofing is an entirely different discussion.
Best Answer
What I've done in the past for mud and other messes is to start with a garden hose to move the mud, using a basic sprayer head. You spray in an arc across the floor and try to get the water & mess flowing in the direction you want. You can use the squeegee broom mentioned by brhans to help with this. The water not only helps liquefy the mess to make it easier to flow away, but the movement and pressure does the work of moving it for you. You don't even need a pressure washer to get most things clean this way.
If you get really good at this, you can not only get large amounts of mud/mess off the floor, but you can get nearly 100% of it off. I've used the technique so much and figured it out so well that mops were essentially unnecessary for several things I've had to clean off floors. Getting the water to move correctly will prevent puddles and will leave not much more standing water than a mop will.
How thick and/or how large an area it is you need to work with determines your starting point. If it's not thick and not a large area, start at the farthest point away from your exit or drain. This gives you the best shot at getting it done in the fewest passes. And don't worry about trying to clean it all in one pass. With something as messy as mud, it'll likely take several passes, even if there's not much there.
If it's a lot of mud/gunk or it's a large area, start closer to the exit or drain. This allows removal of sections of mess and helps prevent it from flowing back on your already cleaned sections.
If it's really thick, as in more than an inch, then you need to shovel it out first, using a flat bladed shovel, like the image below, or a snow shovel or scoop shovel. Depending on where you need to get it, a wheel barrow, like Jim Stewart mentions, could do the trick. If you can't get one through a door, then 5 gallon buckets can work in a pinch.