Our tenant is leaving end of January, and we were thinking of listing it mid Jan, and then taking some time to clean it up etc and having an "open house" for prospective tenants the first weekend of Feb.Any thoughts on this? 

The fire dept turned off the water at the street by the time our guy got there, and the City quickly got involved exerting a fair amount of pressure.In the interests of keeping this story short, these are all the things that we had to do: Communicate with the property owner, the tenants, and the city constantly, offer hotel rooms to affected tenants (some took us up on this), secure the building from future squatters, get the water main and all pipes thawed out, have the water company turn the water on at the street, have a crew of plumbers isolate and repair/replace broken pipes in a muddy crawl space, clean up the mess created by the water in the basement/crawl space, get into all apartments to inspect for damage, test and get water service restored, and repair some internal damage to one of the apartments (that's still in progress), and apologize to the tenants. 

Meaning if they leave on X date and leave the place clean you will not pursue collections on any past due rent.

I saw one that needed work and they literally took a picture like standing on the tub looking down in the bathroom with the toilet open and it was not clean it had something in it….

If they are PVC/ABS look for the clean outs -threaded lids that can be removed- and basically pressure wash the inside of the pipes on all runs.If you are handy, remove the toilette and use high pressure water to clean the pipes, you"ll be surprised with what you can accomplish with a garden hose with a pvc end cap and a couple of drilled holes on it.

She is helping me clean out the stuff downstairs.

Everyone I've used, for renovation, repair, and cleaning, was found by word of mouth.

Before renting it again, clean the place up and then take pictures of everything.

I'm looking for the most accurate way to estimate my expenses.How much can I expect for a cleaning fee?

What you “should” do is personally see that your mothers house is clean, uncluttered and in good repair if you want highest market value.