Hi OP
If you start a thread about an elderly parent, you can trigger people's own issues and attitudes and sometimes get scolded for it.
I'm in a very similar situation to you, not with the Deputyship because I have Lasting Power of Attorney but in many of the other aspects. I'm sorry, I don't know anything further about Deputyships, PP have had more info.
You have done everything right. You have seen that your DM has had all the assessments needed and is going through the system as needed. You are doing this because you want to not because you have a statutory duty to, unlike all these other agencies.
You are very much taking your DM's interests into account, there's no such thing as perfect 'best interests', everything is a compromise. What is very much in your DM's best interest is having a sane and functioning daughter who can advocate in important circumstances and visit her, and in order to do that you need her to be somewhere you can visit without more stress than necessary. You have zero obligation to have her live near you either, and it seems where she currently lives is ideal because she's always lived there and other people could visit too and you yourself can get there, be less stressed by ease of travel and supportive friends in what would otherwise be a lonely situation. I do the same, and I wouldn't listen to random outside opinions on it. Everyone professionally involved with my DM is very supportive of me doing the same as you propose.
I have never had any professional be anything other than grateful for all the admin and money I have sorted out and for the arrangements I've made for my DM. No-one has ever asked me to prove what I am spending money on or making arrangements about. The only time any such thing would come into consideration is when my DM finally runs out of money, on a similar timescale to yours, so I keep decent accounts (there was a period when she had capacity to make broad financial decisions, she doesn't have much idea now). I'm responsible with my DM's money, I make my best judgements that spending on particular things will help her, by far the biggest cost, dwarfing all others, is the care home, and I don't worry about some auditor looking over my shoulder. I do my best to ensure the care home she lives in is fulfilling their role, and others who have a hand in her care. That's a lot. It would not be in my DM's interests for me to hand over POA to her solicitor, stop visiting, wash my hands. I could and no-one would want me to do any of that, so they're happy my needs are met too.
I would not hang on to the house, though, I would let it go when you're able. It's also possible as you know to get an attachment on the property to pay care fees, my DM's local council would have done this. The restrictions and pitfalls of letting, especially being a non-professional absentee landlord, could be very stressful. The income generated would be nowhere near what care costs and you'd likely have to do work to bring it up to regs, not to mention the paperwork, regs, maintenance and angst of bad tenants even if you're paying an agency. I would sell it and lock the money in the highest-interest account possible: interest rates are normalising higher than they used to be. You could get advice from a reputable IFA.
Good luck, get support from people who are on your side.