Care in care homes is very expensive and will eat up all but substantial savings in a year or two.
The sensible approach would be to proceed via Social Services. They will do a care assessment, and in most cases will recommend home carers before they recommend care homes. However, the frequent falling might push your MIL’s needs over the threshold for needing a care home. Don’t dismiss the usefulness of home carers, though: two visits a day from someone who will deal with the incontinence issues can be an absolute godsend.
If your MIL has savings above £23,500, Social Services will require those to be spent first. Once they drop below this figure, Social Services will part-fund the care, whether that is at home or in a home, and once the savings are gone, they will fully fund. That’s why it’s better to go through them than do this independently.
The other service useful for a faller is a pendant or wrist alarm linked to a helpline. This would enable you to go out knowing that you’ll be contacted if she falls.
I know this doesn’t answer your initial question but it’s important to have proper information before broaching the subject.
I would start with “I think this is getting too much for us to cope with on our own” and then see what reaction you get. If your wife isn’t ready to discuss it yet, give her a bit more time. You’ll have planted the seed and she might need some time to work it through in her head. Meantime, do some research about how the care system works. On the other hand, your wife may feel exactly the same as you do and just needs you to start the conversation.
Good luck.