**Hint:** Instead of a spiral, suppose the groove on the LP were made of $800$ concentric circles of equal width; this will make calculations easier and will provide a very accurate approximation.
Since the groove has a non-zero width, the circles aren't really circles; they're annuli). Since you are given the outer and inner radii of the entire track, you can calculate the outer and inner radii of each annulus. Take the average of these to get the radius that the needle will trace, and use $C = 2\pi r$. Then find an efficient way to add up the $800$ lengths.
Here's the calculation for the outermost annulus. First, the width of each annulus will be $$w = \frac1{800}(5.7-2.5).$$ This means that the track inside that annulus will have a radius of $5.7 - \frac w2$, giving a length of $2\pi(5.7-\frac w2)$. Now only $799$ to go...