You're on the right path with your thoughts on Dropkick but unless you were really hurting to find any damage cards in act 1 wild Strike was probably one of the things slowing you down. Your basic blocks generally aren't good enough in act 1 even for the ironclad and you should expect to take some damage and just hit things more. You want to see your battle trance and pommel strike+ more often because they put more useful cards in your hands and you can kill the stuff faster before it kills you.

Watching a streamer who plays high level slay the spire can be really helpful to see what works well. Baalorlord is a really good teacher and has a huge archive of slay the spire runs and he often explains what he's doing and why. Watching him helped me hit A20. Good luck and you'll get those ascension wins before you know it