Dear Suzy
Sounds to me the biggest obstacle to overcome is Mum's reluctant to have help in order to give her a little rest. She probably feels as though she is abandoning Dad if she takes this road, when in fact they will both benefit, particularly as she will feel a little less stressed and more able to spend quality time with Dad.
You know Mum best, but you need a strategy to put this in place. I would arrange a meeting at your local social services with a worker for you and your sister to discuss your concerns. The social worker will understand because she will have encountered this problem many times. Together maybe you can gently coax Mum into the realisation that she cannot go under too because where will Dad be then? There is no shame in needing help, nobody does this job, and it is a labour even if it is of love, 24/7 even the professionals - it can't be done indefinitely.
In additition to the social worker, there is an agency called Crossroads others have mentioned on here, who provide a few hours enabling the carer to get out for a while. Age Concern. Speak to the Alzheimers Society in your area to see what other opportunities there are - all counties vary in the availability of support for carers. I'm sure other members will have other information for you.
Once armed with all of this information gently draw Mum into the idea. But gently does it, best to plant the seed and let her come around to the idea at her own pace.
Take care of yourselves, too
Chesca