Notes

I'll try.
Premise 1) Everything that begins to exist has a cause.
The implied ending here is that everything that begins to exist has a cause for its existence. There is an implied assumption as well...that it is possible for things to begin to exist.
So I would challenge this premise on both implications. First, we don't know anything about the causes of existence. We only know about how things which already exist change; cause is a measure of the change in existing things, not the formation of something from nothing. This is problematic for kalam specifically because we don't know if something from nothing is possible; in fact, most scientists and philosophers agree that this makes no sense. Therefore the mere fact that things do exist refutes the first premise.
Premise 2) The universe began to exist.
This is a scientific unknown, and in fact would violate all we currently know about the universe. From a scientific standpoint it is theoretically impossible. So, much how premise 1 requires the postulation of an impossible event, so does premise 2.
One of the core issues with kalam, and indeed all cosmological arguments, is that they rely on assumptions about the universe to postulate logical truths beyond the universe. But the fact is we don't know anything about how logic or properties work outside of our universe, or even if there is such a thing in the first place in any meaningful sense.
Therefore such things are essentially guesses at best, and concluding a supernatural force to solve an impossible problem is sort of like saying apples are at the end of the set of an infinite number. While I can't say for certain apples are not the end of such a set, having an "end" at all is, as far as known mathematics are concerned, impossible, and if it was possible, there is no way to know whether or not "apples" is the correct answer.
Conclusion: The universe has a cause.
Let's say we accept the first two premises. Premise one states that everything that begins to exist has a cause for its existence. Premise two states that the universe began to exist. Therefore, the universe has a cause.
This is logically valid. The problem is your other claims, which do not follow. These claims are the following:
The cause of the universe is independent of the universe.
The cause of the universe is immaterial/non-physical (same thing).
The cause of the universe lacks space.
The cause of the universe lacks time.
The cause of the universe is supernatural.
The cause of the universe is uncaused.
The cause of the universe has will.
These are problematic for your conclusion. First, there is no guarantee that number one is true; it is possible the universe and its cause are codependent, much how gravity and mass are both the cause of and caused by the existence of the other. Number two is also undetermined by your argument; you have no way of knowing if the cause of the universe was a physical existence outside the universe. Just because everything within our universe is physical does not mean that everything outside it is not. Space and time are also problems because there is no guarantee these properties do not exist in some external reality to our universe, just in a different way. Five is true, but only if you define "nature" as "the universe", which is a pretty broad definition.
Six and seven are the biggest hurdles. Six is completely unsubstantiated; it's entirely possible the cause of the universe was itself caused by something else. As theists are happy to point out, efficient causes are not time-dependent. There could be something beyond, or even lots of somethings beyond, the cause of our universe, that caused it. Seven is a raw rationalization, as natural processes occur regularly with no intentional action; space gas and dust creates stars, but there is no intent built into this. In addition, another assumption I'm sure you're making (call it number eight) is that the cause of the universe is "powerful", but this too is unsubstantiated...just as the dust does not have more "power" than a star, there is no reason to believe the event that caused our universe has any sort of innate power, nor even that it still exists afterwards.
Perhaps you can get to these things with other arguments, but the kalam does not even get you in the ballpark. Hopefully that covered it in a more direct manner, if you have any questions let me know.