One part of my training is striking. One major part of this is the roundhouse. Basically, I kick some striking pads for 3x10 kicks for each leg then repeat, as powerful and fast as possible. I find that this is really difficult. I can do it the first time, and the next time, I can barely finish. I basically exhaust myself in 40 seconds or even less. I believe it's mostly because of my cardio.
How do I increase my endurance for this and prolong the period of sustained power without doing the kicks itself?
Sprint intervals. Kicking is similar to sprinting. 10-15 seconds full sprint with 30-45 seconds rest for 8-12 sets is good. That's what we used to do for tkd back in the old days. That amount of kicking shouldn't be difficult. You can do it.