Explanation:

- The sum of the dimensions (length $l$, breadth $b$, height $h$) of the cuboid is given as 28 cm, i.e., $l+b+h=28$.

- The total surface area formula is $2(lb+bh+lh)=558$. Simplified, this gives $lb+bh+lh=279$.

- We need to find the diagonal ($d$) of the cuboid, found using $\sqrt{l^2+b^2+h^2}$.

- Using formulas, $(l+b+h{)}^2=l^2+b^2+h^2+2(lb+bh+lh)$.

- Replace known values: ${28}^2=l^2+b^2+h^2+2\times 279$.

- Calculate: $784=l^2+b^2+h^2+558$.

- Simplifying: $l^2+b^2+h^2=226$.

- Diagonal $d=\sqrt{226}\approx 15$.

- Correct Answer: Option 4 - 15 cm