I am shocked that they discharged her to home when she was unable to get upstairs to the bathroom etc. There is an emergency provision for up to 6 weeks care from the NHS for patients discharged from hospital. They could / should have provided the commode, visiting care, if needed, walking frame etc. Actually, was she an inpatient? Was she admitted or did she spend the day in A&E? If not admitted the NHS are not required to ensure a care package is in place.
If ongoing care is needed after the NHS 6 weeks, the process is as Luckygirls describes.
Subject to a Care Needs assessment she should be able to get up to 4 visits a day. Carers will help with washing and showering, providing a hot meal, getting dressed / ready for bed . Any toileting needs (e.g dealing with the commode).
This is in addition to Attendance Allowance.
Also check whether your Mum is entitled to Pension Credit.
It is tough. I had the same resistance from my Mum re care coming in, but she is very happy with her cheery carers now.
Start with Adult Services in your Mum's Local Authority.
Age Concern are brilliant at advising you of your Mum's rights to care etc - they put a lot on their web page, and in non-lockdown periods have well trained volunteers who visit and advise on everything you might be entitled to and discuss what might be helpful.