Iron Fist wins here I think.
Check IF's respect thread. You will see an old fight between him and Cap. IF is holding back and Cap isn't,yet it is still a very close fight and IF describes Cap's skill as basic. Now consider that current IF is much much better than he was back then.
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Danny was also near exhausted during that previous fight with Cap. He'd just gotten through smashing the entire Wrecking Crew. Back then, he could only use the fist twice a day. Danny was killing himself trying to use the technique.
And he described it as basic, but the speed&power were overwhelming.
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I think Captain America would win over classic Iron Fist. Current Iron Fist, though.... I've heard he can manipulate chi to heal himself, or use energy attacks, boost his strength/speed/durability to much higher levels, etc...
I don't think Cap could beat current Iron Fist. It just doesn't seem plausible, barring a couple of flukes.
I would actually lean towards Iron Fist here too. Although he kinda did poorly against HYDRA goons/drones earlier on in the The Last Iron Fist Story, whereas I'm pretty sure Cap woulda mopped the floor with em all... his 1v1 fighting abilities are well-suited to take down Cap. Especially since his upgrades.