Concrete – Cleaning indoor concrete floor without damaging it

I just finished helping a colleague clean out his (large) garage at his new house. He's gradually converting it to a indoor gym for himself and his kids (ie: indoor swing set, toys, etc, with a small corner for his weights). The previous owners used the garage for parking their cars, so the floor is a real mess: oil spills, random dirt and other spills, he himself spilled beer and food a few times on it.

We need to clean the floor and make it look presentable, but there's a catch: certain items (bench, wooden play fortress, etc) have already been assembled and bolted into the concrete with anchors. Those areas were already clean, so no worries there, but this means we can't just buy a heavy duty solvent and use a pressure washer afterwards to blast it all out. We're going to have to just mop and scrub.

That being said, is there a particular general purpose cleaner we should use?

• It sounds like the acid-based cleaners wouldn't be a good idea, since they can leave a white film behind and damage the concrete if we don't spray it away with a hose afterwards.
• The concrete also seems to have a waxy finish on it, so I don't want to damage it if it servers an important purpose, so a number of alkaline abrasives also wouldn't necessarily be a good idea.
• We'd considered simple floor cleaner like what one would use on a tile floor, but it doesn't seem to put a dent in the grease/oil stains, and even when diluted, seems to leave a sticky residue behind that takes days to go away. Maybe since the floor is cold, it doesn't evaporate away very quickly.
• We need to use something non-toxic. Kids will be touching the concrete, and we don't want them getting sick if they don't wash their hands before having a snack.

Interesting point: while doing some woodworking, he spilled a lot of carpenters glue on the floor, and when it dried, it easily came off with a paint scraper. The concrete looks very clean, and still has its waxy surface, so there must be a cleaner that can do this if cheap glue can.

Is there some kind of general purpose cleaner I can use to mop the floor with periodically and gradually get out the grease, oil, and other stains. If we have to mop the floor 10 times, that's fine, since he's a bit of a neat freak, and would likely be mopping up footprints from shoes every couple of weeks anyway.

Thank you.