Beauty of this question is not in the reasoning or calculation part, but the three traps it sets up for the people solving. Seldom have I seen so many traps in the same question.

Text solution below, however for those interested in detailed step by step solution (video) along with clear call out of the TRAPS/MISTAKES here is a video solution: Solution for Beginners - 50% Get This Wrong Q464 DS02888 L-Shaped Garden

SOLUTION
I am going to start with statement 2
Statement 2: Perimeter = 60
The perimeter of a polygon is sum of all sides of the polygon. In our case, starting from the bottom and going clockwise:

15 + k + (15-k) + (15-k) + k + 15 = 60

This is TRAP 1: With the constraint of time, some would look at this and say "One equation one variable" and mark this as "Sufficient". But thats the trap if you do not simplify this equation. When you simplify you realize that k cancels off and you only get
60 = 60 !! No value of k possible form the given information.
INSUFF

Statement 1: Area = 189
Pretty easy to come to the quadratic
k^2 - 30k + 189 = 0

TRAP 2: If you stop here and say - obviously k is not unique since 2 possible values of k possible. Hence insufficient. You will be falling into the trap 2. Remember - one of the values of k could be negative or discarded, so you must keep on going here.

k=9 or k=21

TRAP 3: If you stop here and say - well I expected 2 values and I got two "+ve" values of k. No unique answer hence INSUFF! You will be falling into the trap 3 after all the hard work! Just take a step further and check the "feasibility of the two values"

k=9 : possible
k=21: impossible because the max side in the polygon is clearly 15. k cannot be more than that . So we must discard

So only k=9 possible. Hence (A) is sufficient to answer the question

(A)

Check out the video for clear takeaways! Also An EXPERT TIP, chalk this up as one of the most important questions you have to review few days before the exam so you don't fall for these traps.