First, perhaps you noticed that what the others did amounted to (a) a derivation of the formula I referred you to, and (b) a use of the same formula in a different place. There are often multiple ways to approach a problem, which can be scary, but should really be reassuring!
Second, you had a very good start; I presume you marked your drawing with the extra arc measure you found. (Your arithmetic had the right answer, but used a miscopied number so the work you showed was wrong, which misled me.) That is the first step to "cutting clutter": find anything you can immediately derive from what is given, and add it to the picture.
A second step can be, as I suggested in a previous thread, to ignore irrelevant parts. Here, the two radii might be ignored once you've marked the arc as 90 degrees, so you might recopy the figure without them. That can literally remove clutter. (I don't usually do that, because I can't be sure something isn't needed, but it can help.)
The other thing I suggest is that, in addition to seeing what you can do with what is given, you also look at the goal and think about how you might reach it. In this case, it was to find an angle between a chord and a tangent, so your mind should immediately go to the theorem I pointed out. If it doesn't, it might go to one of the ideas the others suggested! The important thing is to have something you are aiming toward. Then take whatever steps will head you in that direction.
I compare this with how you'd find your way back to civilization if someone dropped you in the middle of a wilderness. You'd first check your pockets to find what tools you have that might be useful, and look around to find out where you are and what's available (maybe a tall tree to climb?). Then you'd think about where you are headed -- maybe you know that anyone nearby is more likely to live along a river than on a random mountain top, so you figure you should head downstream. Then you just take it step by step, hoping at each step to discover something new that might help toward the goal. Just keep moving (and keep your mind aware, so you move reasonably).