This work studies the problem of LQG control when the link between the sensor and the controller relies on a Wi-Fi network. Unfortunately, the communication on a wireless medium is sensitive to noise in the transmission band, which is characterized by the Signal-to-Noise Ratio (SNR). Wi-Fi allows to switch among different bit-rates in real-time thus permitting to trade-off lower loss probabilities for larger latency or vice-versa to achieve better closed-loop performance. To exploit this feature, under a constant SNR scenario, we propose a cross-layer approach where the bit-rate is optimally selected based on a control performance metric (i.e. minimum LQG cost) and a model-based controller is used to compensate for the packet losses. Under time-varying SNR, we additionally propose a (sub-optimal) on-line rate adaptation strategy and we guarantee the closed-loop stability under some mild conditions. Numerical comparisons with emulation-based approaches using TrueTime, a realistic Matlab-based Wi-Fi simulator, are included to show the benefits of the adaptive approach under time-varying SNR scenarios.