I was in BA last June/July and on my last day it snowed!!! It was cold, but bearable.
I took Spanish classes at a school, BridgeLingatec, but one of the best teachers at that school gives private lessons, which is what I would recommend if you are both at the same level. Her name is Gabriela Ferrante, http://gabrielaferrante.tripod.com/links.htm. She is also very knowledgable about BA and you can get some good advice from her.
I rented an apt through ByTArgentina, it was very nice for $1000 a month. It was in a noisy location, but very convenient to the school, which is why I chose it. If I were to do it again, I would opt for the Palermo area as it was much quieter and had less of a hectic big city pace.
If by "total immersion" you mean living with a family, it totally depends on your comfort level. It is a great way to learn a language, but can be very stressful as the families usually don't speak any English. I am your age (although I hesitate to admit to being middle-aged) and I did a homestay in Costa Rica for a month. Lovely family, but I found that I need my own space. I found the apartment to be the perfect solution for a long-term stay.
Try to learn some Spanish before you go as it will make those first days all that much easier. Go to your local library and check out a Spanish CD. Watch Spanish TV. Remember, the BA accent is very different than Mexican or Central American accents.
Have a good trip, or as they would say
"Ojala que tengan un buen viaje"