Who Took My Toilet Paper?
J, 66 bytes

1#.2|@-/\1e3&amp;((-:@[%:_1:)^[:i.1+[*0{])((*{:)+[*(1{])*1e3%~i.@#@[)]


Try it online!

This could be golfed more, but I'm putting it away for now.

Instead of taking an analytic approach I am using complex number arithmetic to break the spiral down into 1000 straight line segments per rotation, and then summing those segments.

I find the 500th root of -1, and keep multiplying it by itself to rotate it and get an approximation to the next spiral point.  

Because the spiral also moves outward, we need to take the new vector, normalize it, multiply the normal vector by 1/1000 of the thickness, and then add that little correction to the new vector.

Conceptually, we're doing something similar to approximating a circle's circumference with the short sides of many triangles.



The idea is simple, but the golf part of it boils down to boring bookkeeping and argument parsing which isn't worth going into.  In theory the realization of this method could be a lot shorter.