If you don't mind commuting for 20-30 mins, I've found great hotel options in Insadong area with reasonable prices (50-55 USD per night). I personally prefer hotels for a long stay because of the daily cleaning. I think I'll need it. It's the same subway line (line 3 - orange) so it's a straight way to my clinic, no subway or station change which is nice. Consider TOP hotel, Kuretakeso, Ibis,... They all looks great with a way better price compared to in Gangnam area. I stayed in Gangnam area during my time for consultation. It was quite pricey for a hotel of the same quality.
Edit: Line 3 is for Apgujeong / Sinsa area. The total for my stay was less than USD 600 for 12 nights. Similar private property on Airbnb after fees and taxes are either same or higher. I did not look into hostel so the price point is for hotels/private room or Entire place on Airbnb.