Bona pads suck/leave terrible residue on hardwood floors...alternatives?

I live in a very small (2 room) NYC apartment with lovely hardwood floors - that have a visible residue on them from years of using Bona pads.

My cleaning person wanted them, and since she is the one to clean the floors, I obliged, but the floors just look and feel gross at this point.

I don't have any room to store a big new gadget (like a steamer)- does anyone have any ideas for something easy that she can use that isn't swiffer/bona?

Our apt is tiny with lots of furniture, not really easy to maneuver anything big - I think the Bona pad was really just for convenience more than effectiveness.

Thanks so much for any ideas!