Apply, as the 6 month requirement is that they've had the related condition for 6 months or more. And 'needed help' has a broader understanding than carers, as could be family help (informal care), equipment, telecare or other.
So in the case of someone who is falling at home, and has subsequently had serious fall and broken their hip, it may be that they had been falling before or at high risk of falls, that their mobility has been limited and they had been struggling to bend and complete care tasks, I.e. that you'd already got an alarm in or were checking on them by phone or in person. That they have equipment to assist or /& carers. What's the underlying medical condition? (Osteoporosis, osteoarthriris, Parkinson's, arthritis, chronic back problems, weak knees, issues with blood pressure or significant heart problems? ) Whilst PPs are correct that a fall out of nowhere and broken bones causing care need, on its own would start the 6 months time requirement ! it's not as often that someone falls and breaks their hip without previous medical condition causing increasing disability prior to that.
You need to give them (dwp) the full picture as honestly as you can, which means saying what a bad day looks like. That's very hard for people to write down themselves, so having help with the form (CAB or a professional) is helpful.
In my experience AA assessments by DWP are not like Pip in that they don't require the face to face assessment- whilst they may write to GP, dwp are looking for confirmation of diagnosis and what other professionals are involved in context of what they've been told on form. (So if mum /dad/ relative has seen Occupational therapy & been given/or advised to get equipment some months or years prior- add them onto your list of professionals involved (& ended))
Add the professionals who've seen the person recently too on the form, especially hospital social worker. Or discharge liaison person as they can confirm history of condition.
Good luck, as I hope your relative gets awarded it, if s/he qualifies.