Answer: 720

Small Hint:

Group the chords by the number of vertex steps $k,$ where $k$ is $1,$ $2,$ $\ldots,$ or $6$

Big Hint:

For $k\lt6$ there are $12$ chords of length $24\sin(\frac{k\pi}{12}),$ while there are $6$ diameters

Solution:

For $k$ equal to $1,$ $2,$ $\ldots,$ and $5,$ there are $12$ chords of length $24\sin(\frac{k\pi}{12}),$ and there are $6$ diameters of length $24.$ The five chord lengths are $$\begin{gathered}6(\sqrt6-\sqrt2),\quad12,\quad12\sqrt2,\\12\sqrt3,\quad6(\sqrt6+\sqrt2).\end{gathered}$$ Their sum $U$ is $$\begin{aligned}U&=12+12\sqrt2\\&\quad+12\sqrt3+12\sqrt6.\end{aligned}$$ Therefore the total is $$\begin{aligned}12U+6(24)&=288+144\sqrt2\\&\quad+144\sqrt3\\&\quad+144\sqrt6.\end{aligned}$$ Hence $a=288$ and $b=c=d=144,$ so $a+b+c+d=720.$