It’s understandable that she is angry. She is grieving the body and life and home she used to have. Grief takes the form of anger when the losses are hard to bear. Your mother-in-law spends hours at a time wishing she were home and with people she loves and who she knows loves her. It’s very, very difficult for her to accept that her care is beyond what you can offer and that she has to make the best of being safe but with strangers.
I think you need to enlist the aid of her doctor, especially if she trusts him or her. The doctor can explain to her what is involved in caring for her and why it’s more than you can do at home. You might then also want to work with the social worker at the nursing home or bring in a family therapist to help you all process the grief and find ways to make the transition. She needs help with her feelings of loss and her worry that she is being abandoned. You and your husband need help with any feelings of guilt and regret you may be carrying.
Over-stretched staff at the nursing home aren’t going to be able to give her the attention she deserves to have. Working together with the guidance of a counselor, you, your husband and your mother-in-law may be able to come up with a visitation schedule at the nursing home she can count on. Perhaps she can come to your home for an afternoon or for dinner now and then. If she has a spiritual life, she might enjoy being taken to her house of worship once in a while. If she is unable to go out at all, you may be able to arrange for people she knows to come keep her company or to do some activities with her a couple times a week.
Arguments about why she can’t go home will only cause hurt feelings. Instead, give her lots of reassurance that she is still loved and a schedule of visits and activities to look forward to.
I wish you well.