How do I get mom to understand she needs to bathe and wash clothes?

The internet and the telephone are great tools for researching agencies to help.

I'm not able to do the laundry since I live 3 hours away which also makes it difficult to take her to doctor appointments or find agencies to help.

I'm a big fan of the "Help me help you" approach. I used it with my dad and it worked every time. Make sure your mom knows that her freedom isn't at stake (right now). It's all in the delivery and it may take several discussions before there's a resolution that you can live with. I've told my dad that it hurts me to see him not bathe regularly (yes, guilt). During a personal conversation I asked him if there was a reason he wasn't bathing regularly and if there was, would he share it with me so I could maybe help? He lived with me so he always had clean clothes, you might have to take your mom's clothes and wash them yourself. In my situation I finally hired a bath aide. She came several times and while she got the job done my dad decided that he could actually shower by himself and started doing so. No more visits from the bath aide.

Approach your mom in a non-judgemental way, in a loving way, and in the spirit of wanting what's best for her. And if she still won't bathe tell your mom that you need to hire a bath aide (through an agency) because you don't want her to get sick and she'll feel so much better when she begins to bathe regularly. She may have a reason for not bathing and you just don't know it. Maybe she's embarrassed about something or maybe she has difficulty when she bathes.

But the bathing issue along with the forgetting to pay bills, I would agree that she needs to be seen by another Dr. A specialist or even a neurologist. There are many threads here on this very topic. It might be helpful to you to see what others have done in your situation but caring for someone who is not wanting to bathe is a pretty common problem for caregivers.

Do you have the option of just taking the clothes to the laundromat yourself, or taking them home, washing them then bringing them back? She may not be capable of solving the problem herself anymore (but as you've noticed, able to argue about it NOT being a problem 'til the cows come home). I'm not diagnosing anything , however classic dementia behavior is the arguement of the 10,001 reasons why it is NOT a problem that they started a fire in the kitchen, bounced a check, don't have food in the fridge, haven't bathed, haven't done laundry, etc. etc. etc.

1) It may be time for a new doctor. I am assuming that Mother is not seeing a geriatrician because I cannot imagine a doctor who specializes in the issues of aging ignoring the things you bring up. At this point, Mother's primary care provider should be a geriatrician.
2) I suggest a call to Social Services for a needs assessment appointment. Be present when Mother is interviewed for this. If she is like my mother she will insist that she doesn't need help. You can be the voice of reality. "When was the last time you went to the laundromat? How did you get there? How did you get the clothes home?" Even if Mother will not financially qualify for aid, having an objective party come up with a list of what she needs help with can be a very useful bargaining point in discussing it with her.
3) Assure her that you are on her side. "Mom, I want to see you be able to stay right here as long as you can, and to live as independently as you can. In order to do that you may have to accept a little help in some areas. Isn't it worth putting up with a homemaker coming in twice a week to help with cleaning and laundry so that you can continue living on your own?"

There may be an number of other things she needs help with. Someone should have POA authority and take over bill paying. Maybe meals-on-wheels should be arranged. Does she need help with medications?

If Mom does have dementia, eventually she may not be able to live alone. But if you can get her to see that you are trying to extend the time that that is safe for her and can talk her into accepting a little help to make that possible, it will be a great kindness, even if she fights you on it.