@CalicoPusscat I've just been through all this. DM went into a nursing home a few months ago, sorting out all the financing etc is super complicated but once it's done it's done.
As others have said, if you weren't involved in the placement, social services will have been.
Apologies if you've answered this further up (I couldn't see the answer anywhere) but does she own a property?
Has the care home told you what the weekly cost is?
For DM it's c. £1260 per week but the first 12 weeks were half cost. After that she is full cost because she owns her own flat. If your DM doesn't own a property it will as you say just come down to savings but they will dwindle fast given the costs involved.
Is she in a residential care home or a nursing home? Some places offer both, if so is she receiving nursing care?
If she has been deemed to need nursing care then she will automatically get a contribution of around £225 per week towards care costs (which for us brings the cost down to just £1025 per week.
CHC funding will cover all care costs but is incredibly hard to get. My DM was assessed recently and has very complex needs (advanced Parkinson's and dementia, problematic behaviour, doubly incontinent etc but still didn't get it). You can employ legal firms to help but their support with the process seems to cost about £6k.
If I were you I'd try to have a chat with the Adult Social Care team before you get beneath the funding threshold so that you can have your ducks in a row. But you'll probably need to be very persistent, it's often really hard to get hold of a human to speak to. Very best of luck.
On the plus side, I've found life has been a lot less stressful since DM was in a nursing home. I tried everything to keep her at home but it was absolutely exhausting and things are better for everyone now.