Help me through the coming weeks with my elderly mother.

Be realistic. If you see her around five times a week and do lots for her, she's not really 'very independant' is she? Regardless of age. Quite the opposite.
Time to step away and let the hospital know you are facing carer breakdown and will be doing nothing more. No shopping, cleaning etc. Be also very aware that your DM may likely refuse carers (many don't want to pay) and will tell staff her DD will look after her. They may take her at face value for a quick.discharge.
Agree with pp and be hard as nails. Think long term. This is unsustainable and unfair.

What does your Mum want to do?

If she wants to stay at home there are discharge to assess teams that can put in a short term package of care to assess her at home. Either she will be OK remaining fully independent or they can help transition her to receiving domiciliary care. They can also sort out equipment and adaptations to make her safer.

I understand its a worrying time for you, but you might be jumping the gun somewhat going straight to she needs a care home. There's lots of options before that, none of which have to involve you being primary carer against your will.

If your siblings feel it's necessary suggest they chip in to pay for a live in carer for a few weeks. Reframe every suggestion away from you doing something to If we feel this is needed how do we put it in place with professionals.

The Hospital Social Work team should get involved when she is close to fit to be discharged. Discuss it all with them, but be clear what you can/can't offer. Ultimately though, your Mum has full mental capacity so it's her choice.

This is good advice. I’ve just been through the system with my mum, now in a care home. It’s a long hard road get your hard hat on !

The hospital was ready to discharge my mother back home for my elderly father to take care of her. It wasn't until my brother asked if she was able to walk that they moved her to a rehabilitation unit for a spell and further assessment. Be assertive about her needs.
If she can move about snd weight bear, you could get a care package in to help wash and dress early morning. That has made a huge difference to us. We know someone is there first thing to check, sort breakfast and manage showering. They also leave lunch ready and then we check at the end of the day making sure the house is secure. The early mornings are when the majority of help is needed. My father now accepts help from caters but resisted our pleas to maintain personal care.
At the moment, good carers are helping us keep him in his own home snd taking some of the stress away for us.
On another note, if you don't already have it, get POA in place for health.

This is good advice. The focus of my hospital at every bed meeting is discharge, discharge and discharge. I know of a case pre my NHS days where social services suggested that a young man give up work to care for elderly relative!!

However, to balance this, the longer she is in hospital, the less likely she will ever get back living independently. We see this all the time.

I’m afraid you will have to be brutal - I work in the Care System.

You make it very clear to the hospital that you are NOT available for 24hr care and that you are only able to see her a few times a week on certain days and times. You say your other siblings are NOT available. You say she is not to be discharged from hospital until an appropriate care package is in place. Also, contact SS and reiterate this - put everything in writing via email.

If they think the family will unconditionally support, services will do NOTHING and you will find yourself in an impossible situation and there will be safeguarding concerns for your Mum.

Be hard as nails OP.

Just say it's not a long term solution and they need to look at other more permanent options.
She's 95, realistically her mobility is unlikely to improve at that age.
Once you open the door of becoming the default carer it's very difficult to shut it again