cleaning the basement floor prior to painting it

we have a concrete floored basement that was previously painted. To make a long story short we also have steam heat and the boiler failed. It was replaced amid tons of water, boiler chemicals, dirt, dirt from workmen tracking in and out and a lot of dust from where they moved a device that had been in place for over 25 years. END RESULT is a very dirty floor that needs repainting (some of the boiler chemicals tore off the paint).

So how would you approach this mess? A mop and bucket I don't imagine would be terribly effective. What kind of solvent would leave the floor clean enough to paint? Any ideas at all would help - thanks.