It does take calculus to solve this, but let's see if we can do the work up front to turn this into a formula you can use for any ellipse.
Start with standard form: x^2 / a^2 + y^2 / b^2 = 1, where 2a and 2b are the axes. Let's use the x axis as the major axis. Now, to rotate this θ degrees, we use the transformation
(x,y) -> (x cos θ + y sin θ, -x sin θ + y cos θ)
Plug this in to the ellipse equation:
(x cos θ + y sin θ)^2 / a^2 + (-x sin θ + y cos θ)^2 / b^2 = 1
b^2(x cos θ + y sin θ)^2 + a^2 (-x sin θ + y cos θ)^2 = a^2 b^2
b^2(x^2 cos^2 θ + 2xy cos θ sinθ + y^2 sin^2 θ) + a^2 (x^2 sin^2 θ - 2xy cos θ sinθ + y^2 cos^2 θ) = a^2 b^2
(b^2 cos^2 θ + a^2 sin^2 θ) x^2 + (b^2 - a^2)(2 cos θ sin θ) xy + (b^2 sin^2 θ + a^2 cos^2 θ) y^2 = a^2 b^2
(b^2 cos^2 θ + a^2 sin^2 θ) x^2 + (b^2 - a^2)(2 cos θ sin θ) xy + (b^2 sin^2 θ + a^2 cos^2 θ) y^2 = a^2 b^2
This looks complicated, but for any ellipse where you know the axes and the degree of rotation, this becomes a reasonably simple equation. Let's not plug numbers in just yet, though. First, differentiate, treating a, b, and θ as constants:
2(b^2 cos^2 θ + a^2 sin^2 θ) x + (b^2 - a^2)(2 cos θ sin θ) (xy' + y) + 2(b^2 sin^2 θ + a^2 cos^2 θ) y y' = 0
Set y' to 0:
2(b^2 cos^2 θ + a^2 sin^2 θ) x + (b^2 - a^2)(2 cos θ sin θ)y= 0
y = [-2(b^2 cos^2 θ + a^2 sin^2 θ) / (b^2 - a^2)(2 cos θ sin θ)] x
Since a, b, and θ are constant for a given ellipse, this is a line. It will intersect with the ellipse at the highest and lowest points, where the derivative is zero.
So, for any ellipse, you can use the following steps to find the maximum and minimum without calculus, since we've already done that part of the work.
a = half the major axis = 9.58
b = half the minor axis = 5.74
θ = degree of rotation = 165
Plug into the ellipse equation:
(b^2 cos^2 θ + a^2 sin^2 θ) x^2 + (b^2 - a^2)(2 cos θ sin θ) xy + (b^2 sin^2 θ + a^2 cos^2 θ) y^2 = a^2 b^2
36.8884 x^2 + 29.4144 xy + 87.8356 y^2 = 3023.81
Plug into the line equation:
y = [-2(b^2 cos^2 θ + a^2 sin^2 θ) / (b^2 - a^2)(2 cos θ sin θ)] x
y = -2.50819 x
Substitute y from the line equation into the ellipse equation:
36.8884 x^2 + 29.4144 x*(-2.50819 x) + 87.8356 (-2.50819 x)^2 = 3023.81
515.685 x^2 = 3023.81
x = +/- 2.42150
y = -2.50819 * (+/- 2.42150) = -/+ 6.07358
We have found two points: (2.42140, -6.07358) and (-2.42140, 6.07358). These are the highest and lowest points on an ellipse that is the same size and orientation as yours, but centered on the origin. All you need to do now is add your center coordinates to shift these to their proper location.
(2.42140 + 148.281, -6.07358 + 19.2309) = (145.860, 25.3045)
(-2.42140 + 148.281, 6.07358 + 19.2309) = (150.703, 13.1573)
Edit: I did a little more algebra; here's a different method you may find easier to use.
The ellipse equation (b^2 cos^2 θ + a^2 sin^2 θ) x^2 + (b^2 - a^2)(2 cos θ sin θ) xy + (b^2 sin^2 θ + a^2 cos^2 θ) y^2 = a^2 b^2
can be rewritten as
A x^2 + B xy + C y^2 = F
where
A = b^2 cos^2 θ + a^2 sin^2 θ
B = (b^2 - a^2)(2 cos θ sin θ)
C = b^2 sin^2 θ + a^2 cos^2 θ
F =a^2 b^2
Now looking at the line equation, it turns out that this is
y = (-2A / B) x
Substituting for y gives
A x^2 + B (-2A / B) x^2 + C(-2A/B)^2x^2 = F
which can be rearranged to
x = +- B sqrt(F / (4A^2 C - AB^2) )
And therefore
y = -+ 2A*(sqrt(F / (4A^2 C - AB^2) )
So, for any given ellipse, the steps become:
1. Find A, B, C, and F using the equations above.
2. Find K = sqrt(F / (4A^2 C - AB^2)
3. x = +- BK, and y = -+ 2AK
4. Add your center coordinates to the x and y calculated in step 3.