Algorithm for a 2D path-finding puzzle
One approach would be to bring out the big hammer and try using a SAT solver.  You introduce boolean variables $x_{\ell,p,o}$, which is true if piece $p$ is placed in orientation $o$ in location $\ell$ on the grid.  Then, you write down boolean clauses to encode the conditions for a grid to be valid.  Finally, you do binary search: given $k$, you try to find a placement that has score at least $k$ (by writing boolean clauses to force the score to be at least $k$ and then using a SAT solver to test whether there exists an assignment to the variables that satisfies all of the clauses), and then do binary search on $k$.  This might be effective, or it might be slow; it's hard to predict, so the only way to find out is to try it.

The trickiest part of encoding into SAT will probably be calculating the score.  I suspect it will be useful to introduce extra boolean variables $y_{\ell}$, which is true if the piece at location $\ell$ is connected to the spool.  Then, write constraints that capture the relationship between the $x$'s and $y$'s.