Have you tried Murphy's Oil Soap?
Bona works great on hardwood for cleaning. Spray, wipe down, then buff while slightly damp.
I mop mine with 1/2 cup apple cider vinegar, 1 Tbsp blue Dawn, 2 gallons water. Wring mop out thoroughly. It will dry rather quickly and leave the wood shiny as new. It there is a residue, it may take a few times to get them back to original shine. I've been using this since 2001 & mine still look like new.
It is probably varnished or varathaned, so all you would need is a damp cloth dipped in warm water and a birt of dish liquid.
After you've cleaned them with your normal cleaning solution, give them a " final rinse " by wiping them down w/a mix of vinegar & hot water. A cup or 2 of vinegar mixed into a bucketful of water will do the trick.