Definitely best to do one miter cut first, then put it in position just outside the wall and you can draw the line for the opposite end by running a pencil along the part it will butt up against. Also best to "measure twice, cut once". The pencil might end up marking a wee bit short, depending on how thick it is and how easy it is to get it and your hand where they need to be to do the marking.
This way, if the first cut turns out to be not quite right, you haven't already cut the board to length and can redo that one. And better to cut the second end a bit long than a bit short because you can always cut a bit more off. BUT: this isn't furniture. If the brace ends up half an inch short, just screw it in as it fits. It will still serve the function.
You can buy a miter box for not too much money (or even make your own), you don't NEED a sliding compound miter saw. Good excuse to get one though, you'll find all sorts of uses for it as time goes by. "Honey, I'd love to make this coop for your chickens but I'll need a sliding compound miter saw to do it"
Also note, as per Gargoyle's picture that there is a RIGHT way and a WRONG way to put the brace in a a gate or door. The brace NEEDS to go from the top corner farthest from the hinge side to the bottom corner on the hinge side. I brings the cantilevered weight of the door back to the hinge which is attached (we hope) to a very strong support.
I'm assuming that when you say it is shaky, you mean side to side, not toward and away from the coop. Unless I'm mis-reading the pictures the studs on the long walls are a lot closer together than the coop is wide. For rigidity, the studs should really be 90 degrees from what you have, and nailed to a 4x4 or 2x4 on the flat. It gives the base of the stud a larger "foot print" to lever against side to side. Bracing the end wall will help stabilize the long wall, at least to the first stud. Again, ASSUMING
you don't want diagonal bracing across the width of the coop on the inside, you can help those closer to the coop by adding diagonal bracing to the roof. One diagonal per roof "section" will help a lot. There is no shape that wants to deform less than a triangle.
Oh, and be careful, though Gargoyle said the miter cuts should be 45 degrees, that is ONLY if the space you are bracing is a perfect square. You CAN use 45 degrees, but the brace won't be able to go corner to corner. That's why I use the mark and cut method. It is a lot easier than calculating out the angles.
Bruce
This way, if the first cut turns out to be not quite right, you haven't already cut the board to length and can redo that one. And better to cut the second end a bit long than a bit short because you can always cut a bit more off. BUT: this isn't furniture. If the brace ends up half an inch short, just screw it in as it fits. It will still serve the function.
You can buy a miter box for not too much money (or even make your own), you don't NEED a sliding compound miter saw. Good excuse to get one though, you'll find all sorts of uses for it as time goes by. "Honey, I'd love to make this coop for your chickens but I'll need a sliding compound miter saw to do it"

Also note, as per Gargoyle's picture that there is a RIGHT way and a WRONG way to put the brace in a a gate or door. The brace NEEDS to go from the top corner farthest from the hinge side to the bottom corner on the hinge side. I brings the cantilevered weight of the door back to the hinge which is attached (we hope) to a very strong support.
I'm assuming that when you say it is shaky, you mean side to side, not toward and away from the coop. Unless I'm mis-reading the pictures the studs on the long walls are a lot closer together than the coop is wide. For rigidity, the studs should really be 90 degrees from what you have, and nailed to a 4x4 or 2x4 on the flat. It gives the base of the stud a larger "foot print" to lever against side to side. Bracing the end wall will help stabilize the long wall, at least to the first stud. Again, ASSUMING

Oh, and be careful, though Gargoyle said the miter cuts should be 45 degrees, that is ONLY if the space you are bracing is a perfect square. You CAN use 45 degrees, but the brace won't be able to go corner to corner. That's why I use the mark and cut method. It is a lot easier than calculating out the angles.
Bruce