The policlínico of the Hospital Aleman has Otorrinos (ENT specialists) available without appointments. I've seen a couple there over the years, and all are excellent. All of the doctors I've met there speak at least some English.
Dr. Juan Pablo Inda did a residency in the US and speaks better English than I do. In fact he likes to see angloparlantes to keep his language skills up. A few years ago he helped me recover from exactly the same symptoms you describe. If you call the hospital, you can ask for his schedule in the policlínico and then go there when he's on duty. If by chance you're assigned to someone else, just tell the doctor that you want to see Inda, and they'll arrange it.
I have also seen Dr. Di Giacinti at the policlinco - he's also very good, but his English is only adequate.
If you've never been to the policlínico, the drill is simple. You enter at the south side of the building, on Pueyrredón by Beruti. Show your passport and a credit card at one of the windows on the right as you enter, and they will charge you a fixed fee and give you a form showing a number. When your doctor is available - usually no more than 20 minutes later - your number will appear on an electronic call board along with the number of the doctor's consultorio.