It is always hard once the dementia gets to the stage when it is no longer safe for them to be on their own. I wasnt able to look after my mum in my home either so she had to move to a care home. She too had tried to make me promise not to put her in a care home, fought tooth and nail against going and begged me to let her come and live with her.
Im afraid it is par for the course for the person with dementia to demand to go home to begin with and 3 weeks is nothing at all. It took mum 6 weeks and this is quite quickly - its usually a couple of months or more. Once mum had settled she thrived there. She was eating properly and getting her medication so she was physically better than she had been for ages. her paranoia went, she made friends and joined in the activities and she liked having someone around at all times of the night and day so she wasnt scared any more.
What is your mum like when you are not there? Quite often the PWD is quite happy in the care home, but seeing relatives can trigger this behaviour. If you are upset when you visit you could be inadvertently reinforcing this - I used to find that mums mood mirrored mine, so if I was upset she would be too, but if I was happy she would be. Go in with a bright smile stapled firmly to your face and as soon as you see that she is getting upset distract her - I used to take in little treats for this very purpose. I never found that comforting her made things better, in fact it just used to make it build up and build up, but if I could distract her before it started I could prevent it.
Reasoning with her did nothing either as she was completely unable to understand why she was there and trying to explain made her angry and the whole thing just got worse. I would simply say that I was sorry she was there, but she was convalescing and she could go home when the doctor said - then change the subject. If, despite all of this she got very upset I would take myself "off to the loo" and call over one of the care staff. I also used to visit her in the lounge and not go upstairs to her room until further down the line once she had settled. Being in a public place meant that she had other things to distract her and that I could talk about, without her endlessly mulling over her grievances.
Please dont cry in front of her as she will find this really distressing and make her frightened. If you feel the mask slipping take yourself "off to the loo"
Dont stay long and dont do long drawn out goodbyes; I didnt say goodbye at all as this would often make her want to come with me. I used to leave my coat and bag in the managers office (other people on here leave them in the car) so there was no visual cue that I was going. I would say I needed to talk to the manager and that I would be "back soon". I also used to time my visits so that it was in the mornings when she was at her best - dont go afternoon/evening because of sundowning - and just before the mid-day meal so that she had something to distract her when I left. If she gets very upset be prepared to cut your visit shorter than you had intended.
Afterwards, remind yourself that she is warm, well fed, cared for and - above all - safe. Perhaps you could plan to do something nice for yourself afterwards. I used to go to a nearby cafe for lunch. Finally, please please ditch the guilt. It is not your fault that her dementia has progressed and she now needs a whole team of people giving 24/7 support.