I have never understood the cult following that the Gerloczy enjoys. So admittedly I am biased.
I wouldnt trust any hotel that guarantees they will run their a/c in late September when the outside temps are in the 40's, 50's and 60's or even low 70's. European air conditioning is miserable to begin with and most places are kept warm by U.S. Standards. At least the K&K was honest. If you want a near guarantee of 68F in your room you might look at renting an apartment.
The K&K is a large company, my experience is that their facilities including a/c and elevators are in good repair and modern. The front desk staff is also very accommodating and helpful, or so has been my experience in the 3 or 4 times we stayed there. My one experience with the
Gerloczy wasn't so good.
For location, the K&K is right at the Opera entrance to the M1 metro. This admittedly is my favorite part of town and after about 6 years of traveling to Budapest we purchased our vacation home in the same general area. From the K&K a short ride on the metro one stop either direction takes you to one of 2 tram lines that reaches 75% of the sight seeing. But you will need to walk up and down the steps to the metro stop. That's about 24 steps down.
The Gerloczy seems closers but will require a little more walking to reach the right transportation lines..
Because of your limitations you might want to check out the Carat Botique Hotel on Kiraly u. Or the Roombach they are both very close to Deak Ferenc ter which is a major metro intersection making reaching the sights convenient.
If you need a taxi my favorite is City Taxi at +36 1 211 1111; but unless your condition is really bad the M1 Metro and the Trams reach most things. The trams are like HOHO Busses but cheaper and no climbing. The M1 metro is history in and of itself. Not like any you will ride elsewhere; its the oldest on the continent of Europe.