I have hardwood oak floor in kitchen and dinning area, I use white vinegar and water, yes water, but don’t use a mop, I use a damp not wet sponge in the vinegar and water solution, it is a get down on you knees job, then I apply a coat of orange glo for hardwood floors really puts a shine on the floor. Depending on the traffic in the area where the floors are you can run a dust mop over the floors and sweep up any excess dirt, dust etc. I clean mine once a month and do the orange glo for hardwood as these areas of the house get alot of use. Make sure not to track in any stones in your shoes, we usually take our shoes off and put on slippers when we come in the house. You may want to get a humidifier since your area is dry.
MURPHY’S OIL SOAP. It not only cleans, it shines them as well.