I don’t want this to be a case of too many cooks, but I will add my couple of ingredients to the soup.
I’m going to agree with @mabloodhound here and leave the fan and its light and two switches alone. That part of it has been working fine for a long time and I see the box on the fan as being a suitable J-box and the lag screws into the support structure as being adequate.
Now the third switch you have. It is likely controlling half of one of the wall outlets would be my guess and would be easy to figure out using a lamp as a tester. If that proves out then you can decide if you want that or not and that can be fixed by unhooking that wire at the outlet and putting in a jumper on the outlet, then pulling that wire back up to the attic space to be the feed for your new lights. If it was set up to feed the whole outlet then you could leave that end alone and connect the switch end into a full time power source in the box and run a new wire from that switch up the wall and into the attic to feed the chain of new LED fixtures. The third option is that switch controls something else or nothing at all. In both cases you need to follow that wire and see where it goes. I found wires in my old house that ran into the attic and were wire nutted off and hidden in the insulation. So anything is possible and you want to figure it out.
After all that is done then you can work on actually selecting some LED fixtures and figuring out how to hook them up.
I’m going to agree with @mabloodhound here and leave the fan and its light and two switches alone. That part of it has been working fine for a long time and I see the box on the fan as being a suitable J-box and the lag screws into the support structure as being adequate.
Now the third switch you have. It is likely controlling half of one of the wall outlets would be my guess and would be easy to figure out using a lamp as a tester. If that proves out then you can decide if you want that or not and that can be fixed by unhooking that wire at the outlet and putting in a jumper on the outlet, then pulling that wire back up to the attic space to be the feed for your new lights. If it was set up to feed the whole outlet then you could leave that end alone and connect the switch end into a full time power source in the box and run a new wire from that switch up the wall and into the attic to feed the chain of new LED fixtures. The third option is that switch controls something else or nothing at all. In both cases you need to follow that wire and see where it goes. I found wires in my old house that ran into the attic and were wire nutted off and hidden in the insulation. So anything is possible and you want to figure it out.
After all that is done then you can work on actually selecting some LED fixtures and figuring out how to hook them up.