Most of the very very good nail salons here could handle ingrown toenails
The one I go to in Lotte mall has excellent proper trained staff, not just a girl with a nail file and "you wan gel only twenny dolla"
So you could try a decent salon and see what they can do and maybe they could at least do a temporary fix and trim the nail and put some stuff on it to fix it, the iodine, vitamin etc
All the equipment is clean, new, sterilised etc
Otherwise I guess just google or try at hospital? I don't have personal experience but my salon manages my elephant feet and can also do spa treatments for dry skin and what not