Consider a distributed control problem with a communication channel connecting the observer of a linear stochastic system to the controller. The goal of the controller is to minimize a quadratic cost function in the state variables and control signal, known as the linear quadratic regulator (LQR). We study the fundamental tradeoff between the communication rate r bits/sec and the limsup of the expected cost b. In the companion paper, which can be read independently of the current one, we show a lower bound on a certain cost function, which quantifies the minimum mutual information between the channel input and output, given the past, that is compatible with a target LQR cost. The bound applies as long as the system noise has a probability density function, and it holds for a general class of codes that can take full advantage of the memory of the data observed so far and that are not constrained to have any particular structure. In this paper, we prove that the bound can be approached by a simple variable-length lattice quantization scheme, as long as the system noise satisfies a smoothness condition. The quantization scheme only quantizes the innovation, that is, the difference between the controller's belief about the current state and the encoder's state estimate. Our proof technique leverages some recent results on nonasymptotic high resolution vector quantization.