In this paper, the $G^2$ interpolation by Pythagorean-hodograph (PH) quintic curves in $\mathbb{R}^d$, $d ≥2$, is considered. The obtained results turn out as a useful tool in practical applications. Independently of the dimension $d$, they supply a $G^2$ quintic PH spline that locally interpolates two points, two tangent directions and two curvature vectors at these points. The interpolation problem considered is reduced to a system of two polynomial equations involving only tangent lengths of the interpolating curve as unknowns. Although several solutions might exist, the way to obtain the most promising one is suggested based on a thorough asymptotic analysis of the smooth data case. The numerical algorithm traces this solution from a particular set of data to the general case by a homotopy continuation method. Numerical examples confirm the efficiency of the proposed method.