How to deal with a toxic employee when HR is useless and person is all buddy buddy with them (it’s also a small business)

I’m hoping someone out there can give me some advice on how to deal with a toxic person at work. Ok so here’s there situation. I am a supervisor and I’ve had “Sue” on my team for 8 months. I’ve been in the role 1 yr 2 months. She gets in everyone’s business. Seems to enjoy knowing everything. She is blunt and rude. Eg. Our HR Manager was having a little joke with her about something and she told her to shut up. Now yes it was said in jest tone but I don’t know I was brought up to respect the business hierarchy and not to speak out of line. Another example of her behaviour is deciding when she wants to greet myself as well as the State Sales Manager in the mornings when she walks in. It’s blatant disrespect, forget hierarchy for a moment but I think as a human being if I greet acperson the morning you kind of expect reply. When I ask her a work related questions she ignores me as if I’m not talking or she will give a split second glance then will start doing something else as if what I’m saying doesn’t matter it shits me to tears. So I’ve dealt with this behavior for 6 months. Then one day I had had just about enough and I was talking to her and she gave me the glance and got up to look at something in a colleagues drawer and proceeded to start chatting to her while I was talking. I let out a deep sigh out of pure frustration. To which she replied I was listening. I said I’m not getting in to this now ( the owners wife was in that day!!) so I had to maintain composure because I was going to loose my shit. So Sue went down to HR had something to say, came back picked up her bags and left. Said nothing. So that was that. That afternoon my boss called me in to find out what happened so I explained it to him. I admitted I probably shouldn’t have let out the sigh but along with what I’ve mentioned her and so many more inappropriate behavior etc. The next day she came back to work and obviously tension was thick. So a mediation was called. So HR and my manager asked us to talk. She sat all quiet. I was so angry. So my leg started twitching from having to compose myself. I asked her what the problem is to which she replied I don’t know. I said well you walked out so what happened yesterday. So she starts crying and says work is her happy place and I STRESS HER OUT BECAUSE SHE NEVER KNOWS WHAT MOOD IM GOING TO BE IN BECAUSE I BRING MY HOME LIFE TO WORK!! She says she also has issues but she doesn’t blurt them out to work. I was gob smacked. It was a blatant LIE! I shook my head and said I hammering know what to say to this. Then she pipes up Oh, and YOU DONT GREET ME IN THE MORNINGS!!!! another lie. I said to her I’m sorry if you feel me talking about my family JUST LIKE YOU AND EVERYONE ELSE DOES. I always greet you and if you feel I haven’t greeted you sometimes but perhaps I never heard ( knowing ful well I have Always greeted her) unless on the phone I wave or look and smile. Then she brought up an email I sent out that we had ALREADY DISCUSSED AND RESOLVED. And that’s exactly what I said to her again. It’s like she was clutching at straws. I then was asked what would I like her to improve on as her supervisor. I just asked that she look at me when I’m asking her a question and acknowledge me just a person too. She agreed. I walked out the “mediation” frustrated, angry, like seething angry. My integrity was being challenged because she was out right lying and played the victim. But I smiled and said ok all good.....As did she. The following week was sombre. Minimal conversation, strictly work It was just the two of us. By Friday I asked her if we could have a chat about some of the things she said tome in the mediation.Like I’m moody and I bring my home life to work etc. and said what happened? How did we get here? she waved her hand and said oh please forget I said that, I DIDNT MEAN IT! and she goes honestly I have no idea how we got here. Let’s just move on. In my head I’m going ‘ok so she’s completely white lying to throw you under the bus but wtf. Don’t LIE like that. I told her we really need to work together. She “agreed “. A couple days later she had a day off for a specialist appt. that day I was asked where she is and I said she gone for an appointment. Then a few others turn round saying “no, she’s gone to watch her daughters sports carnival.” I honestly would not have cared less what the reason for her leave was but why lie to me and tell other people otherwise. She CONSTANTLY texts and uses her phone. Despite being told by me and MY manager. I’m talking overuse to the point she has to charge her phone by about 2pm! No joke. Now I have spoken to my manager at length about this but he tells me he can’t do much because she works well. Which she does what needs to be done but no more. If I ask her to do something she’ll do it but with attitude So I’m morally conflicted because I feel she is trying to push my buttons to make me loose my shit. But I love my job and will not leave. She’s now all buddy buddy with the apparent HR. So I feel now powerless because the HR opinion is biased and she’s lying to the right people but then “apologizing” to me. It’s such a messed up situation. What do I do? How do I deal with the person? Is a small/medium business so news travels bloody fast 😒