You will need a penicillin based AB for mastitis, but I have also had to use Noroclav as one of my girls seems to get mastitis that doesn't react to penicillin (but that gets expensive to find out...).
I would stay clear of teat tubes, where you have to put the AB straight up the teat, I have had good results with intramuscular injection.
Have the kids been on a bottle at all? If she has triplets I would top them up with bottles anyway as they rarely have enough milk to feed three as well as the kids need it (about 2 ltrs per kid per day). If the nanny is amenable to being milked that may be your best option - as after mastitis the milk yield inevitably goes down and if the kids are still fairly young they need topping up with milk form other goats or Lamlac. She will be reluctant to let the kids back onto the bad side anyway, and you will have to strip that side out at least once a day from now on, otherwise mastitis is likely to recur quite quickly.