I've recently got into road cycling and I'm not particularly strong yet but I absolutely love climbing.
However, I'm very short and thin (5'7" and 130 pounds) and on climbs that average 9%, my cadence is about 60 (at threshold) whilst my usual cadence is about 100-105. After 3km, my legs are completely cooked.
I'm running a 54-39 (or 53?) and 11-28 at the back.

I'm tossing up if I should be getting a compact chainring or semi compact, or if I just need to train harder and stop being a sook.

I'm aiming to do some climbs that have 1km+ of 20% but at least 50% of my training is on flat.

Any suggestions on what I should change to?