Sucking the life out of me. When is it time for a Nursing Home?

I’m sorry to post this, since so many have it so much harder. My mother is 81 and she sits in front of a tv all day, just the sound makes me crazy! She showers herself sometimes she needs to coaxed, she uses the toilet but dripples a lot and doesn’t smell the urine. Her sheets smell like urine and I try and change them 2 times a week but she won’t let me, she lies and says she took a small walk but it’s for 2 minutes!! She can fix small meals when she wants to but won’t grocery shop , I work 12 hours nights 3 days a week, I’m a nurse. My sister lives with us and helps on the weekends, during the week she works day shift 4 days a week comes home and drinks beer in her room and shuts the door! I have a 4 year old granddaughter and a new granddaughter on the way in April. I try and take her with me to my daughters but it’s getting so busy and all she wants is something to eat. She doesn’t try to do anything and her mind is somewhat sharp, she does have frontal ischemic infarcts that are old. I’m getting so angry at her I’ve been doing this for 5 years and have become depressed and have no excitement for life anymore, my brother will pick her up once every few months for a day, no other family around. she has no friends I took her to a adult daycare and she hated it and refused to go back. I threaten to put her in a nursing home and she says go ahead because she can’t understand what she’s doing wrong, she thinks it’s normal to watch TV day and she’s not hurting anyone. Nursing homes are too expensive she only gets social security. Am I wrong feeling like this?  I literally feel the life getting sucked out of me. I do yoga, meet with friends once a week . When is it time for a Nursing Home? Thank you for listening to me.