How do you handle taking care of someone who is 91 and work a nine hour day also?

Not possible to do and stay sane. There is help in the community and I would suggest that you seek some really good advice before you completely burn out. This is not a win-win for anyone. Call you local Area Agency on Aging (these agencies are in every state and provide great information) and help both of you out of a very difficult and non-productive situation. One last bit of advice start today!

You realize your limitations and call in an aide or find a nursing home.

I think that by asking this question you're aware that you're trying to do something you can't. What are you looking for in the way of permission or advice? You know you need help. I am confident that you will find the kind of help that fits your situation.

I've known very spry, engaged 90+ year-olds and I've also seen folks who, despite extraordinary efforts to reach them, are literally just waiting for their ticket to be punched and simply want their physical needs taken care of in the meantime. So, it depends upon the individual.
Full-time home care is very expensive (generally much more costly than assisted living and unaffordable for most people) and, for many, isolating. If were a 91 year-old capable of socializing in any way, I'd like to have the company of others, even if just to watch the action.
Also, if someone needs a lot of care, once the full-time aide goes home for the day, who provides the care? The loved one who just came home from a full day's work.
It's absolutely true that checking your parent into an assisted living community doesn't leave you footloose and fancy-free. You're still going to be running errands, managing finances, spending lots of time there, and of course, being on call. However, it does mean you can at least put your feet up at the end of your work day, have a good night's sleep, and enjoy the company of your friends and family on the weekends. Also, many find they are able to have a renewed and more pleasant relationship with their parent.

You know, nursing homes are not for everyone! People are so quick to say that, even assisted living takes time and money. But if they are not absolutely needing to go, then they should stay at home. It costs a bit of money, but I called a company who came in my home. I had them come from 7:30 until 4 pm so I could work, run errands. It cost 20.50 an hour. They would read to her, bathe her and fix her meals. Beware of agencies that are "locators" I found these places just wanted your email address so they could send endless junk. I was lucky to have a friend refer them to me. They were privately owned and the even helped with light chores. That's when the old fashioned phone book works best.
But simply "put them in a nursing home" is not easy, not necessary until the end, and very costly. Plus, do people realize that before the elderly can go anywhere - they need to be assessed first by the facility? Good luck.

Look for a Senior Center in your area. I started bringing Mom to one and it has been a HUGE relief for me and fun for her. She has severe dementia.

Depending on how much care this person needs they may be eligible for some services through Medicare or Medicaid or even the VA. If this person is a Veteran or the widow of a Veteran contact your local Veteran's service office and find out if they are eligible for some " Aide and Attendance or a Veterans Disability Pension to help with the costs of having someone come in during the day when you are gone to work.

I did this by using a medical adult daycare. I hired someone for 2 hours mornings to get mom up, dressed and to daycare as I had to be at work by 7am. I got out of work and picked her up every day right afterwards. After 4 years I then left my job to take care of Mom fulltime because money ran out and its SO difficult. Its much easier working outside the home, it keeps you sane and gives you more energy to spend the evening and weekends with your Mom. Good Luck!