Mom (87) lonely?

My sister and I live with our 87 year old mom. She is in fair health, can eat, wash and dress herself. She does not drive. We have a very small family and most are gone already. Mom has two friends she only talks to via phone in frequently. We take her out occasionally for lunches and dinners and to atlantic city every once and a while. We take her grocery shopping and to doctor's appointments. Other than that mom pretty much stays in door and watches tv, reads the newspaper and plays computer games. We prepare most dinner meals almost every nite of the week, but once a week, we do our own thing and we don't cook and she either eats a frozen meal or makes her own dinner, which she is capable of doing. Lately, mom is becoming extremely depressed about not going out enough and not having any friends and when we say we will do our own thing for dinner, she yells and starts acting up. I work and my sister is theday care provider for our mom. She does all the house stuff and the shopping and errands. I work and on the weekend help out with the cleaning and cooking. We are both tired and lacking time to ourselves, which is not really the issue. The issue is that mom is complaining more and more about being alone. Our father passed 10 years ago and ever since mom has been steadily going downhill emotionally. We have tried to get her to go to adult day care during the day where there are people her age she can talk to and spend time doing things. But she flatly refuses. We just can't be mom's social activity all of the time. We are exhausted just from keeping things going and me working and my sister dealing with mom day in and day out. We don't know how to help mom. And this does not seem like a medical issue but more an emotional or mental issue (not that mom has dementia, she does not). She just is tired of being alone most of the time. We don't know what to do. She is very lonely and we are very exhausted. We just can't be there socially for all the time. I know that sounds awful, but it is just the truth. What can we do to help her. And when she starts up she gets abusive and uses dirty language and forgot to tell you the most important thing ... that is that mom has never really been a nice person (it pains me to say). I guess you might say she is what you folks call narsacisstic (if I'm spelling this right!). She can be most hurtful and nasty and she is normally not too much fun to be around. We spend as much time as we can with her but we also need our rest. What options do we have for mom?