Is there anyone who gives seniors a bath?

Home health care sent a very pleasant CNA twice a week. Her only job was to give baths, and she traveled from house to house in her own car.

It is often easier to hire someone from the outside to come in with the task of bathing your loved one, as your loved one typically won't give the outside help as much grief as they would a family member, however that being said most agencies do require at least a 4 hour commitment.
JoAnn29 had some great suggestions below, but on top of all that I will add that you can also buy some extra large body wipes and the waterless shampoo and conditioner caps to use on her in between her monthly bath. They work great. I had to use them on my husband occasionally, and the shampoo caps I not only used for him but also for myself when I had my hip replaced and couldn't get in the shower for a while. You can order both the wipes and shampoo caps on either Amazon or Walmart.com.
And the nice thing with the wipes is that you can use them on her while she is lying down in bed.
I wish you the best in getting this issue figured out.

Agencies have people that will do showers or bathing of the client.
Many like to have a set number of hours booked 2 hours for a bath that would include getting the person undressed, into shower or bath, dressed again and change bedding if needed.

I have bathed someone. Nice you have a slide chair. The soap being liquid or a cake to me does not make a difference other than the puff ball makes more soap and I guess the puff helps exfoliate.

I have always personally had hand held shower heads. So if you don't have one get one. Very easy to install. The bathroom should be warm so in Winter heat up with a little heater if bathroom runs cold. Summer, turn off a/c to that room. There are magnetic covers you can put over the vent.

Rinse Mom off first with warm water. Then suds her up real well in a nooks and crannies. Rinse her off. Washing hair maybe the challenge. Mom was Ok with bending her head backward for a rinse and me sudsing her up then I rinsed her hair. I had a towel nearby to wrap around her head. To get to the private area I had her stand with her legs apart holding onto a rail. I washed her rectum area with soap but not her urethra area. (Soap in the urethra can cause a UTI) I used the showerhead to clean the urethra area and rinsed the soap from the rectum area. Make sure a towel is nearby and dry her quickly. I had Moms clothes nearby to get her dressed in the warmth of the bathroom.

Look into hiring a caregiver with experience who is willing to come in weekly to help mom bathe. I'm sure there will be a minimum 4 hour charge or something equivalent, so prepare yourself.