I’m overwhelmed. Any suggestions?

First...NEVER try to pick someone up. You can hurt yourself or hurt them. Call 911 and ask for a "Lift Assist" 99.9 % of the time there will be no charge if there is no transport to the hospital.
She should at least be in Rehab to recover from the broken hip (if it is recent)
And if it was recent even more reason that you should have called 911.
If the surgery for the hip was recent there is a possibility that the confusion might be from anesthesia and the hospital stay.
As far as the bath goes, taking a bath or shower is frightening. The possibility that there would be a slip and fall of the wet floor, if she is confused there are many steps to taking a shower or bath and she may have forgotten some of the steps.
Taking care of someone with a fractured hip can be difficult.
Taking care of someone with dementia is difficult.
Combine the two and it is REAL hard.
Your mom might be better off in Rehab for a while then assess if coming home is possible. Skilled Nursing facility might be safer for her and for you.

It sounds like your mom should be in a nursing home. Why isn't she?

Sounds like this situation is way too much for you. I am wondering why mom did not go to rehab after hospital for her broken hip.

I hate to say it but she needs either 24/7 care at home or move her into a nursing home.

Nice sister, what is she doing to help? My guess is probably nothing.

Call your mom's primary and he can connect you with a social worker to get some info about placing mom. In the meantime they can schedule a home health nurse to check in and an aide to clean mom.

You desperately need help like yesterday!

Its certainly not your fault! Elderly fall, break things, and start to lose mental edge. Sometimes people can manage to care for an elderly parent and sometimes its too much for them. Sometimes the elder needs 24/7 supervision, hygiene assistance, etc etc that is simply too much for an individual to manage alone. You may or may not be able to get in-home assistance depending on her financial situation. Placing an elder in a nursing home is not a failure or lack on your part. Its placing the elder in a situation that will best meet their current needs. (and sometimes there are ALOT of needs)